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train | 0.44.7 | \subsection{The Superpropagator}
The semiclassical expressions for $F_a^{'}(t)$ and $F_a^{''}(t)$
allows us to solve the equations of motion (\ref{eq22}) and
(\ref{eq23}) and to calculate the superpropagator,
Eq.(\ref{superprop}). In order to have explicit formula, we shall
consider only the zero order approximation $F_a^{''}(t) \approx
\phi_{xx}(t)/2$. Substituting it in (\ref{eq22}) and (\ref{eq23})
and integrating by parts we get
\begin{eqnarray}
\ddot{r}_e(t)+\Omega^2_0\, r_e(t)+\frac{\gamma^2}{m}\int_0^t\mathrm{d}s
F(t-s)\dot{r}_e(s)&=&-\frac{\gamma^2}{m}r_e(0)F(t), \label{eq-3}\\
\ddot{y}_e(t)+\chi^2_0\, y_e(t)-\frac{\gamma^2}{m}\int_0^t\mathrm{d}s
F(t-s)\dot{y}_e(s)&=&\frac{\gamma^2}{m}y_e(0)F(t), \label{eq-4}
\end{eqnarray}
where $\Omega_0^2=\omega_0^2-\gamma^2F(0)/m$, $\chi^2_0=\omega_0^2+
\gamma^2F(0)/m$ and
\begin{eqnarray}
F(t)=\int\mathrm{d}s\phi_{xx}(t-s),\label{eq-5}
\end{eqnarray}
as in the classical case (eq.(\ref{eq17})).
In Appendix B we solve these equations by the method of Laplace
transforms. We find that the solutions involve two very different
time scales. The shortest time scale is relevant only for times much
smaller than $1/\omega_0$, the period of the decoupled oscillator.
For times of the order of $1/\omega_0$ these terms can be discarded
as transients. Here we shall adopt this approximation and keep only
the terms that are significant for times of $1/\omega_0$. In this
case we show that (\ref{eq-3}) and (\ref{eq-4}) can be written as
\begin{eqnarray}
\ddot{r}_e(t)+2\Lambda\dot{r}_e(t)+\Omega_0^2r_e(t)=0, \quad
\Omega_0^2\gg\Lambda^2
,\label{eq-6} \\
\ddot{y}_e(t)-2\Lambda\dot{y}_e(t)+\chi_0^2y_e(t)=0,\quad
\chi_0^2\gg\Lambda^2, \label{eq-7}
\end{eqnarray}
where
\begin{eqnarray}
\Lambda=\frac{\gamma^2}{2m}\lim_{t\to\infty}\int_0^{t}\mathrm{d}s
F(t-s).\label{eq-8}
\end{eqnarray}
Approximating $\Omega_0 \approx \chi_0 \approx \omega_0$ we find
\begin{eqnarray}
r_e(t)=e^{-\Lambda
t}\left\{\frac{\sin{[\omega_0(T-t)]}}{\sin{(\omega_0 T)}}r(0)+
e^{\Lambda T} \frac{\sin{(\omega_0 t)}}{\sin{(\omega_0
T)}}r(T)\right\},\nonumber \\
y_e(t)=e^{\Lambda
t}\left\{\frac{\sin{[\omega_0(T-t)]}}{\sin{(\omega_0 T)}}y(0)+
e^{-\Lambda T} \frac{\sin{(\omega_0 t)}}{\sin{(\omega_0
T)}}y(T)\right\}.\label{eq-10}
\end{eqnarray}
Within this approximation we can calculate the real part $\tilde{S}$
of the effective action, Eq.(\ref{sefreal}). We obtain
\begin{eqnarray}
\tilde{S}[r_e,y_e]&=&[m\omega_0^2
K(T)-m\Lambda]r(T)y(T)+[m\omega_0^2 K(T)
+m\Lambda]r(0)y(0) \nonumber \\
& &-m\omega_0^2L(T)r(0)y(T)-m\omega_0^2N(T)
r(T)y(0),\label{eq-13}
\end{eqnarray}
where,
\begin{eqnarray}
K(T)&=&\frac{1}{\omega_0}\frac{\cos(\omega_0 T)}
{\sin(\omega_0 T)}, \label{eq-14}\\
L(T)&=&\frac{1}{\omega_0}\frac{e^{-\Lambda T}}{\sin(\omega_0 T)},
\label{eq-15}\\
N(T)&=&\frac{1}{\omega_0}\frac{e^{\Lambda T}}{\sin(\omega_0 T)}.\label{eq-16}
\end{eqnarray}
For the imaginary part $\phi$ of the effective action,
Eq.(\ref{sefimag}), we get
\begin{eqnarray}
\phi[y_e,y_e]=\gamma^2\int_0^T\mathrm{d}t\,y_e(t)\int_0^t\mathrm{d}s\,
F_a^{'}(t-s)y_e(s).\label{eq-17}
\end{eqnarray}
Using again only the zero order term of the semiclassical expression
(\ref{eq49}) for $F_a^{'}(t)$ and the fact that the classical
correlation function decays exponentially, as in (\ref{eq-8}), we
can approximate, for $t \sim 1/\omega_0$
\begin{eqnarray}
\int_0^t\mathrm{d}s\,F_a^{'}(t-s)y_e(s)\approx y_e(t)
\lim_{t\to\infty}\int_0^{t}\mathrm{d}s\,F_a^{'}(t-s) \equiv y_e(t)
\frac{B'}{\hbar}, \label{eq-20}
\end{eqnarray}
which implies that
\begin{equation}
\begin{array}{ll}
\phi[y_e,y_e] &= \displaystyle{\frac{\gamma^2B'}{\hbar}}
\int_0^T\mathrm{d}t\,y^2_e(t) \\
&=\displaystyle{\frac{\gamma^2B'}{\hbar}}[A(T)y^2(T)+B(T)y(T)y(0) +
C(T)y^2(0)],\label{eq-22}
\end{array}
\end{equation}
where
\begin{eqnarray}
A(T)&=&\frac{[-\omega_0^2e^{-2\Lambda
T}+(\Lambda^2+\omega_0^2)-\Lambda^2\cos{(2\omega_0
T)-\Lambda\omega_0\sin{(2\omega_0
T)}}]}{4\Lambda(\omega_0^2+\Lambda^2)\sin^2{(\omega_0 T)}},
\label{eq-x1} \\
B(T)&=&\frac{[-\omega_0^2\sinh{(\Lambda
T)}\cos{(\omega_0 T)}+\Lambda\omega_0\cosh{(\Lambda T)}\sin{(\omega_0
T)}]}{\Lambda(\omega_0^2+\Lambda^2)\sin^2{(\omega_0 T)}},
\label{eq-x2} \\
C(T)&=&\frac{[\omega_0^2e^{2\Lambda
T}-(\Lambda^2+\omega_0^2)+\Lambda^2\cos{(2\omega_0
T)-\Lambda\omega_0\sin{(2\omega_0
T)}}]}{4\Lambda(\omega_0^2+\Lambda^2)\sin^2{(\omega_0 T)}}.
\label{eq-x3}
\end{eqnarray}
Finally, putting everything together we get
\begin{eqnarray}
\lefteqn{J(r(T),y(T),r(0),y(0))=G(T,0)\exp{\left\{\frac{i}{\hbar}\tilde{S}
[r_e,y_e]\right\}}\exp{\left\{-\frac{1}{\hbar}\phi[y_e,y_e]\right\}}
}\nonumber \\
&=&G(T,0)\exp{\left\{\frac{i}{\hbar}\tilde{K}_2(T)r(T)y(T)\right\}}
\exp{\left\{-\frac{1}{\hbar}\tilde{A}(T)y^2(T)\right\}}\times \nonumber \\
&
&\exp{\left\{\frac{i}{\hbar}\left[\tilde{K}_1(T)r(0)y(0)-\tilde{L}(T)r(0)y(T)
-\tilde{N}(T)r(T)y(0)\right]\right\}}\times \nonumber \\
& &\exp{\left\{-\frac{1}{\hbar}\tilde{B}(T)y(T)y(0)-\frac{1}{\hbar}
\tilde{C}(T)y^2(0)\right\}} \label{eq-24}
\end{eqnarray}
where
\begin{eqnarray}
\tilde{K}_{1,2}(T)=m\omega_0^2K(T)\pm m\Lambda
\qquad&\tilde{L}(T)=m\omega_0^2L(T)&\qquad
\tilde{N}(T)=m\omega_0^2N(T) \nonumber \\ \nonumber \\
\tilde{A}(T)=\frac{\gamma^2B'}{\hbar}A(T)\qquad&\tilde{B}(T)=
\frac{\gamma^2B'}{\hbar}B(T)&\qquad\tilde{C}(T)=\frac{\gamma^2B'}
{\hbar}C(T). \nonumber \\
\label{eq-23}
\end{eqnarray}
We finish this section with a comment about the physical situation
described by these calculations. Since we have considered only the
first terms of the semiclassical expressions for $F_a^{'}$ and
$F_a^{''}$ our results are valid only in the Ehrenfest time scale of
the chaotic system given by \cite{beenaker}
\begin{eqnarray}
t_E\sim\frac{1}{\lambda_L}\ln{\left(\frac{S_c}{\hbar}\right)},
\end{eqnarray}
where $\lambda_L$ is the Lyapunov exponent and $S_c$ is a typical
action of the chaotic system, for example the action of the shortest
periodic orbit. Approximating $\lambda_L$ by $\alpha$, we see that,
in order to observe effects for $t\sim 1/\omega_0$, we must have
\begin{eqnarray}
t_E\sim\frac{1}{\omega_0}=\frac{1}{\alpha}\ln{\left(\frac{S_c}{\hbar}\right)}
\Rightarrow
\frac{S_c}{\hbar}=\exp{\left(\frac{\alpha}{\omega_0}\right)}.
\label{eq-25}
\end{eqnarray}
For NS, $\alpha/\omega_0\sim 8$ and $S_c \approx 10$ which means
that $\hbar$ must be smaller than $10^{-3}$ for our results to be
valid. | 2,852 | 28,405 | en |
train | 0.44.8 | \section{Applications}
The superpropagator allows us to study the time evolution of the
oscillator under the influence of the chaotic system. The reduced
density matrix satisfies
\begin{eqnarray}
\rho_o(r(T),y(T))=\int\mathrm{d}r(0)\mathrm{d}y(0)\,J(r(T),y(T),r(0),y(0))
\,\rho_o(r(0),y(0)). \label{eq-26}
\end{eqnarray}
In the following we calculate explicitly the propagation of two
different oscillator's states. These two applications are similar to
the ones presented by Caldeira and Leggett to study dissipation and
decoherence.
\subsection{Propagation of a Gaussian State.}
For a Gaussian state
\begin{eqnarray}
\psi(z(0))=\frac{1}{(2\pi\sigma^2)^{1/4}}\,e^{\frac{i}{\hbar}pz(0)/\hbar}
e^{-z^2(0)/4\sigma^2} \label{eq-28}
\end{eqnarray}
the density matrix $\rho_o(z(0),z'(0))=\psi^*(z'(0))\psi(z(0))$ can
be written in terms of $r=(z+z')/2$ and $y=z-z'$ as
\begin{eqnarray}
\rho_o(r(0),y(0))=\frac{1}{(2\pi\sigma^2)^{1/2}}\,e^{\frac{i}{\hbar}py(0)}
e^{-r^2(0)/2\sigma^2}e^{-y^2(0)/8\sigma^2}. \label{eq-29}
\end{eqnarray}
Substituting (\ref{eq-24}) and (\ref{eq-29}) in (\ref{eq-26}) and
performing the integrals, we get
\begin{eqnarray}
\lefteqn{\rho_o(r(T),y(T))=G(T,0)\left[\frac{2\pi\hbar^2}{2\hbar
\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T)}\right]^{1/2}}\nonumber \\
&\times \exp{\left\{-
\frac{\tilde{N}^2(T)}{2[2\hbar\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T)]}
\left(r(T)-\frac{p}{\tilde{N}(T)}\right)^2\right\}}&
\nonumber \\
&\times\exp{\left\{
-\left[\frac{\tilde{A}(T)}{\hbar}+\frac{\sigma^2
\tilde{L}^2(T)}{2\hbar^2}-\frac{(\sigma^2\tilde{K}_1(T)\tilde{L}(T)-\hbar
\tilde{B}(T))^2}{2\hbar^2[2\hbar\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T)]}
\right]y^2(T)\right\}}& \nonumber \\
&\times\exp\left\{\frac{i}{\hbar}\tilde{K}_2(T)r(T)y(T)
-\frac{i}{\hbar}\frac{(\sigma^2\tilde{K}_1(T)\tilde{L}(T)
-\hbar\tilde{B}(T))}{(2\hbar\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T))}
\tilde{N}(T)
\left(r(T)-\frac{p}{\tilde{N}(T)}\right)y(T)\right\}&, \nonumber \\
\label{eq-30}
\end{eqnarray}
where,
\begin{eqnarray}
\tilde{C}_1(T)=\tilde{C}(T)+\frac{\hbar}{8\sigma^2}. \label{eq-31}
\end{eqnarray}
For $y=z-z'=0$, $\rho_o$ becomes the probability density. After
normalizing we obtain
\begin{eqnarray}
\rho_o(r(T),0)&=&\left[\frac{\tilde{N}^2(T)}{2\pi[2\hbar\tilde{C}_1(T)+
\sigma^2\tilde{K}_1^2(T)]}\right]^{1/2}\nonumber \\
&\times&\exp{\left\{-\frac{\tilde{N}^2(T)}
{2[2\hbar\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T)]}\left(r(T)-\frac{p}
{\tilde{N}(T)}\right)^2\right\}}. \label{eq-33}
\end{eqnarray}
Eq.(\ref{eq-33}) represents a Gaussian packet whose center follows
the trajectory
\begin{eqnarray}
r(T)=\frac{p}{\tilde{N}(T)}=\frac{p}{m\omega_0}e^{-\Lambda
T}\sin{(\omega_0 T)}. \label{eq-34}
\end{eqnarray}
The dissipative effect due to the interaction with the chaotic
system is explicit. The same behavior was obtained by Caldeira and
Leggett \cite{caldeira1} using a thermal bath with many degrees of
freedom. Eq.(\ref{eq-34}) represents the trajectory of a weakly
damped harmonic oscillator. The critical and strongly damped cases
cannot be described by this formalism because of the weak coupling
regime adopted.
The width of the evolved packet is given by
\begin{eqnarray}
\sigma^2(T)&=&\frac{\sigma^2\tilde{K}_1^2(T)+2\hbar\tilde{C}_1(T)}
{\tilde{N}^2(T)} .
\end{eqnarray}
After some algebra we can show that
\begin{eqnarray}
\sigma^2(T)&=& \sigma^2\bigg\{\frac{(1+\epsilon^2)e^{-2\Lambda
T}}{1+ \epsilon^2} \nonumber \\
&+&\frac{\Gamma[1-e^{-2\Lambda T}(1+2\epsilon\sin{(\omega_0
T)}\cos{(\omega_0 T)}+2\epsilon^2 \sin^2{(\omega_0
T)})]}{1+\epsilon^2}\bigg\}, \label{eq-35}
\end{eqnarray}
where
\begin{eqnarray}
\epsilon=\frac{\Lambda}{\omega_0}\qquad\mathrm{and}\qquad\Gamma=\frac{E_c(0)}
{\hbar\omega_0}. \label{eq-36}
\end{eqnarray}
The expression above for $\Gamma$ comes from the following
considerations: from $\tilde{C}_1(T)$ it follows that
\begin{eqnarray}
\Gamma=\frac{\gamma^2B'}{\hbar m\omega_0\Lambda}.
\end{eqnarray}
Using Eqs.(\ref{eq-8}) and (\ref{eq17}) and the relation
\cite{weiss}
\begin{eqnarray}
\langle p_x(0)x(t)\rangle_e=-\frac{\partial}{\partial t}\langle
x(0)x(t)\rangle_e
\end{eqnarray}
we obtain
\begin{eqnarray}
\Lambda=\frac{\gamma^2 B'}{m E_c(0)},
\end{eqnarray}
which leads directly to (\ref{eq-36}). Notice that, due to
(\ref{eq-25}), $\Gamma\gg 1$ is the only possibility.
Fig.3 shows that $\sigma^2(T)$ for $\Gamma=1$, $0.5$ and $2.0$.
These curves can be well fitted by the simpler expression
\begin{eqnarray}
\sigma^2(T)=\sigma^2[e^{-2\Lambda T}+\Gamma(1-e^{-2\Lambda T})],
\label{eq-37}
\end{eqnarray}
which, for $t\sim 1/\omega_0$, can be written as
\begin{eqnarray}
\sigma^2(T)=\sigma^2[1+(\Gamma-1)2\Lambda T]. \label{eq-38}
\end{eqnarray}
We see that $\sigma^2(\Gamma-1)\Lambda$ plays the role of a
diffusion constant. Fig.3 also shows that $\Gamma$ controls the
increase or decrease of $\sigma^2(T)$. In the present case,
$\sigma^2(T)$ can only increase because of the constraint $\Gamma\gg
1$. In the Caldeira-Leggett model, on the other hand, the width can
also decrease if the the temperature is very low.
\subsection{Superposition of Two Gaussian States.}
We now consider an initial state consisting of two Gaussian
wave-packets, one at the origin and one centered at $z(0)=q_0$:
\begin{eqnarray}
\psi(z(0))&=&N^{1/2}[\psi_1(z(0))+\psi_2(z(0))]\nonumber \\
&=&N^{1/2}\left\{\exp{\left[
-\frac{z^2(0)}{4\sigma^2}\right]}+\exp{\left[-\frac{(z(0)-q_0)^2}
{4\sigma^2}\right]}\right\}. \label{eq-39}
\end{eqnarray}
The density matrix is given by
\begin{eqnarray}
\rho_o(z(0),z'(0))=&N&[\rho_{11}(z(0),z'(0))+\rho_{22}(z(0),z'(0))\nonumber \\
&+&\rho_{12}(z(0),z'(0))+\rho_{21}(z(0),z'(0))]. \label{eq-40}
\end{eqnarray}
with $\hat{\rho}_{ij}=|\psi_i\rangle\langle\psi_j|$. The time
evolution of $\rho_o$ can again be calculated with Eq.(\ref{eq-26}).
The result, for $y=z-z'=0$ is
\begin{eqnarray}
\rho_{11}(r(T),0)&=&\frac{1}{2[1+h(T)]}\left(\frac{\tilde{N}^2(T)}
{\pi\tilde{f}(T)}\right)^{1/2}
\exp{\left\{-\frac{\tilde{N}^2(T)}{\tilde{f}(T)}r^2(T)\right\}},
\label{eq-41} \\
\rho_{22}(r(T),0)&=&\frac{1}{2[1+h(T)]}\left(\frac{\tilde{N}^2(T)}
{\pi\tilde{f}(T)}\right)^{1/2}
\exp{\left\{-\frac{\tilde{N}^2(T)}{\tilde{f}(T)}\left[r(T)-Q(T)\right]^2
\right\}}, \label{eq-42} \\
\rho_{12}(r(T),0)&+&\rho_{21}(r(T),0) \, = \nonumber \\
& &\frac{1}{2[1+h(T)]}\left(
\frac{\tilde{N}^2(T)}{\pi\tilde{f}(T)}\right)^{1/2}\exp{
\left[-\frac{q^2_0}{8\sigma^2}g(T)\right]}\exp{\left\{-\frac
{\tilde{N}^2(T)}{\tilde{f}(T)}r^2(T)\right\}} \nonumber \\
&\times&\exp{\left\{-\frac{\tilde{N}^2(T)}{\tilde{f}(T)}\left[r(T)-Q(T)
\right]^2\right\}}\nonumber \\
&\times&2\cos\left\{\frac{\hbar\tilde{N}^2(T)}
{4\sigma^2\tilde{f}(T)\tilde{K}_1(T)}
\left[\left(r(T)-Q(T)\right)^2-r^2(T)\right]\right\}, \label{eq-43}
\end{eqnarray}
where
\begin{eqnarray}
\tilde{f}(T)&=&2[2\hbar\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T)],
\label{eq-44} \\
h(T)&=&\exp{\left\{-\frac{q^2_0}{8\sigma^2}\left[1+g(T)\right]\right\}},
\label{eq-45} \\
Q(T)&=&\frac{\tilde{K}_1(T)}{\tilde{N}(T)}q_0, \label{eq-46} \\
g(T)&=&\frac{2\hbar\tilde{C}(T)}{2\hbar\tilde{C}_1(T)+\sigma^2
\tilde{K}_1^2(T)}.
\label{eq-47}
\end{eqnarray}
The interference term can also be rewritten as
\begin{eqnarray}
\rho_{12}(r(T),0)&+&\rho_{21}(r(T),0)=2\cos{\left[a(T)((r(T)-Q(T))^2-r^2(T))
\right]}\nonumber \\
&\times&\rho_1^{1/2}(r(T),0)\rho_2^{1/2}(r(T),0)\exp{\left[-\frac{q^2_0}
{8\sigma^2}g(T)\right]}. \label{eq-48}
\end{eqnarray}
Eq.(\ref{eq-48}) shows that the interference is attenuated by
$\exp{[-(q^2_0/8\sigma^2)g(T)]}$. Eq.(\ref{eq-48}) is very similar
to the expression obtained by Caldeira and Leggett \cite{caldeira2},
although there is no temperature dependence in $g(T)$, which can be
written as
\begin{eqnarray}
g(T)=\frac{\Gamma\,b(T)}{(1+\epsilon^2)+\Gamma\,b(T)}, \label{eq-49}
\end{eqnarray}
with
\begin{eqnarray}
b(T)=e^{2\Lambda T}-1-2\epsilon\sin{(\omega_0 T)}\cos{(\omega_0
T)}-2\epsilon^2\sin^2{(\omega_0 T)} \label{eq-50}
\end{eqnarray}
and $\epsilon=\Lambda/\omega_0$. We note that the asymptotic limits
\begin{eqnarray}
g(T=0)=0\qquad\mathrm{and}\qquad g(T\rightarrow\infty)\rightarrow 1
\label{eq-51}
\end{eqnarray}
are the same as those in the Caldeira-Leggett model.
Fig.4 shows $g(T)$ for $\Gamma=10$. In the regime $\Gamma\gg 1$, we can
approximate $g(T)$ by
\begin{eqnarray}
g(T)=\frac{2\Gamma\Lambda T}{1+2\Gamma\Lambda T}. \label{eq-52}
\end{eqnarray}
This simplified expression helps to estimate of the decoherence
time. For example, with (\ref{eq-52}), we can estimate the time $T'$
such that
\begin{eqnarray}
\exp{\left[-\frac{q^2_0}{8\sigma^2}g(T')\right]}\sim
10^{-3}. \label{eq-53}
\end{eqnarray}
Defining $n\equiv q^2_0/8\sigma^2$ (the number of quanta
$\hbar\omega_0$ of the wave packet centered at $q_0$), we get
\begin{eqnarray}
\left[\frac{n-\ln(10)}{3\ln(10)}\right]2\Gamma\Lambda
T'=2\tilde{n}\Gamma\Lambda T'=1\Rightarrow
T'=\frac{1}{2\tilde{n}\Gamma\Lambda}. \label{eq-54}
\end{eqnarray}
Since we are interested in the situation where $n \gg 1$ and $\Gamma
\gg 1$, we find that the decoherence time is much smaller than the
time scale where dissipation takes place, i.e., $T' \ll 1/\Lambda$. | 3,990 | 28,405 | en |
train | 0.44.9 | \section{Discussion and Conclusions}
We have made two important assumptions in our calculation of the
superpropagator. The first of these assumptions, the weak coupling
regime, was important to reduce the path integral to a quadratic
form in the oscillator variables. The second assumption was the
semiclassical regime of the chaotic system. This was essential to
establish the connection between the coupling in the influence
functional and the classical correlation and response functions that
enter in the classical description of the system. The use of these
classical functions make the importance of the chaotic dynamics
explicit and show that the time scales obtained classically are
important ingredients to describe dissipation. In particular, the
exponential decay of correlations happens in a time scale much
shorter than the natural period of the oscillator. The time of
correlation loss plays the role of the microscopic time scale in the
Brownian motion, which is much shorter than the macroscopic one
\cite{reif}. Moreover, the exponential decay of the classical
correlations is what makes dissipation possible in the present
treatment. The corrections due to periodic orbits have not been
explored here and the importance of their contribution to
dissipation and decoherence is not clear at this point.
The effective dynamics we obtained, expressed in (\ref{eq-24}), is
analogous to the Caldeira-Leggett theory in the limit of high
temperatures and weak damping \cite{caldeira1,caldeira2}. For
example, the diffusion constant in (\ref{eq-38}) can be written,
for $\Gamma\gg1$, as
\begin{eqnarray}
\sigma^2\Gamma\Lambda=\frac{E_c(0)}{2m\omega_0^2}\Lambda,
\end{eqnarray}
which should be compared with
\begin{eqnarray}
D=\frac{k_B T}{m\omega_0^2}\Lambda
\end{eqnarray}
for the Brownian motion. Therefore, $E_c(0)$ plays the role of $k_B
T$. From Fig.4, $\Gamma$ seems to play the role of $k_B
T/\hbar\omega_k$ since it controls the behavior of $\sigma^2(T)$.
However, despite this close analogy between the two models, our
results are valid only for short times since they are limited by
Ehrenfest time and perturbation theory.
In summary, we have shown, using Feynman-Vernon approach, that a
chaotic system with two degrees of freedom can induce dissipation
and decoherence in a simple quantum system when weakly coupled to
it. The formalism we have chosen allows us a close analogy with the
many body formulation of the Caldeira-Leggett model. The most
important quantities in the formalism, the correlation and response
functions, are obtained directly from the dynamics, and not from
phenomenological assumptions as in the Caldeira-Legget model. In our
approach we have used simple classical approximations and discarded
all periodic orbits corrections. The effects of these corrections
are certainly worth studying.
\begin{appendix}
\section{The Stationary Phase Approximation}
In this appendix we solve the path integral Eq.(\ref{pathint}) by
the stationary phase approximation. Let $(r_e(t),y_e(t))$ be the
stationary path and
\begin{eqnarray}
&r(t)=r_e(t)+\delta r(t)=r_e(t)+\epsilon_1\tilde{r}(t)& \\
&y(t)=y_e(t)+\delta y(t)=y_e(t)+\epsilon_2\tilde{y}(t)&,
\end{eqnarray}
be a neighboring path with $\tilde{r}(T)=\tilde{r}(0)=0$ and
$\tilde{y}(T)=\tilde{y}(0)=0$.
The stationary path is obtained from the condition
\begin{equation}
\begin{array}{ll}
\Delta \tilde{S} & \equiv
\tilde{S}[r_e(t)+\epsilon_1\tilde{r}(t),y_e(t)+\epsilon_2
\tilde{y}(t)]-\tilde{S}[r_e(t),y_e(t)] \\
&= \epsilon_1\frac{d\Delta\tilde{S}}{d\epsilon_1}+\epsilon_2\frac{d\Delta
\tilde{S}}{d\epsilon_2} = 0
\end{array}
\end{equation}
We find
\begin{eqnarray}
\frac{d\Delta\tilde{S}}{d\epsilon_1}
=-\int_0^T\mathrm{d}t\tilde{r}(t)\left\{m[\ddot{y}_e(t)+\omega_0^2
y_e(t)]-2\gamma^2\int_0^t\mathrm{d}sF_a{''}(s-t)y_e(s)\right\},
\end{eqnarray}
and
\begin{eqnarray}
\frac{d\Delta\tilde{S}}{d\epsilon_2}
=-\int_0^T\mathrm{d}t\,\tilde{y}(t)\left\{m[\ddot{r}_e(t)+\omega_0^2
r_e(t)]-2\gamma^2\int_0^t\mathrm{d}sF_a{''}(t-s)r_e(s)\right\},
\end{eqnarray}
where we used $\tilde{r}(T)=\tilde{r}(0)=0$,
$\tilde{y}(T)=\tilde{y}(0)=0$ and
$\int_0^T\mathrm{d}t\int_0^T\mathrm{d}s=2\int_0^T\mathrm{d}t\int_0^t
\mathrm{d}s$.
Therefore, the equations of motion for the stationary path are given
by
\begin{equation}
\ddot{y}_e(t)+\omega_0^2y_e(t)-\frac{2\gamma^2}{m}
\int_0^t\mathrm{d}sF_a{''}(s-t) y_e(s)=0\label{yclas}
\end{equation}
and
\begin{equation}
\ddot{r}_e(t)+\omega_0^2r_e(t)-\frac{2\gamma^2}{m}
\int_0^t\mathrm{d}sF_a{''}(t-s) r_e(s)=0.\label{rclas}
\end{equation}
Expanding $\phi$, Eq.\ref{sefimag}, around the stationary path we
find
\begin{eqnarray}
\lefteqn{\phi[r_e(t)+\epsilon_1\tilde{r}(t),y_e(t)+\epsilon_2
\tilde{y}(t)]=} \nonumber \\
& &=\frac{1}{2}\gamma^2\int_0^T\mathrm{d}t\int_0^T\mathrm{d}s[y_e(t)+
\epsilon_2
\tilde{y}(t)][y_e(s)+\epsilon_2\tilde{y}(s)]F_a{'}(t-s) \nonumber \\
& &=\frac{1}{2}\gamma^2\int_0^T\mathrm{d}t\int_0^T\mathrm{d}sy_e(t)y_e(s)
F_a{'}(t-s) + \nonumber \\
& & \; \frac{1}{2} \gamma^2
\epsilon_2\int_0^T\mathrm{d}t\int_0^T\mathrm{d}s
[\tilde{y}(t)y_e(t)+y_e(s)\tilde{y}(s)]F_a^{'}(t-s) + \nonumber \\
& & \; \epsilon_2^2 \frac{1}{2} \gamma^2
\int_0^T\mathrm{d}t\int_0^T\mathrm{d}s
\tilde{y}(t)\tilde{y}(s)F_a^{'}(t-s)=\phi[y_e,y_e]+2\varphi[\tilde{y},
y_e]+\varphi[\tilde{y},\tilde{y}].
\end{eqnarray}
Therefore, from (\ref{pathint}), we have
\begin{eqnarray}
\lefteqn{J(r(T),y(T),r(0),y(0))=} \nonumber \\
& &e^{\frac{i}{\hbar}\tilde{S}[r_e,y_e]}
e^{-\frac{1}{\hbar}\phi[y_e,y_e]}\int_0^0\mathrm{D}\delta y(t)
\mathrm{D}\delta r(t) e^{\frac{i}{\hbar}\tilde{S}[\delta r,\delta y]}
e^{-\frac{2}{\hbar}\varphi[\delta y,y_e]}e^{-\frac{1}{\hbar}
\varphi[\delta y,\delta y]}. \label{superpro2}
\end{eqnarray}
We are now going to show that (\ref{superpro2}) is a function of the
initial and final times only, which is not obvious because of the
functional dependence on $y_e(t)$. In order to do this we discretize
the paths and re-write (\ref{superpro2}) in the form
\cite{feynman2}:
\begin{eqnarray}
\lefteqn{\exp\left\{\frac{i}{\hbar}\tilde{S}[\delta r,\delta
y]\right\}\approx} \nonumber \\
& &\exp\bigg\{\frac{i}{\hbar}\bigg[\sum_{j=1}^N \epsilon \,m \bigg(\frac
{(\delta r_j-\delta r_{j-1})(\delta y_j-\delta y_{j-1})}{\epsilon^2}
-\omega_0^2\delta r_{j-1}\delta y_{j-1}\bigg) \nonumber \\
& &+\gamma^2\epsilon^2\sum_{j=1}^N\sum_{k=1}^N \delta y_j\delta r_k
F_{a_{(j-k)}}^{''}\bigg]\bigg\},
\end{eqnarray}
where $\delta r_j=\delta r(t_j)$,
$F_{a_{(j-k)}}^{''}=F_a^{''}(t_j-t_k)$,
\begin{eqnarray}
\exp\left\{-\frac{1}{\hbar}\varphi[\delta y,\delta y]\right\}\approx
\exp\left\{-\frac{1}{\hbar}\sum_{j=1}^N
2\gamma^2\epsilon^2\sum_{k=1} ^N \delta y_j\delta y_k
F_{a_{(j-k)}}^{'}\right\}
\end{eqnarray}
and
\begin{eqnarray}
\exp\left\{-\frac{2}{\hbar}\varphi[\delta y,y_e]\right\}\approx
\exp\left\{-\frac{1}{\hbar}\sum_{j=1}^N
4\gamma^2\epsilon^2\sum_{k=1} ^N \delta y_j y_{e_{k}}
F_{a_{(j-k)}}^{'}\right\}.
\end{eqnarray}
Grouping the exponents we obtain
\begin{eqnarray}
\exp{\left\{\frac{i}{\hbar}\tilde{S}[\delta r,\delta y]
-\frac{1}{\hbar}\varphi[\delta y,\delta y]
-\frac{2}{\hbar}\varphi[\delta y,y_e]\right\}} \approx
\exp{\left\{-\frac{i}{2}U^T M U -A^T U \right\}},
\end{eqnarray}
with
\begin{eqnarray}
U^T\equiv(\delta r_1 \ldots \delta r_N \, \delta y_1 \ldots \delta y_N)
\qquad M\equiv \left( \begin{array}{cc}
0 & p \\
p & r
\end{array} \right),
\end{eqnarray}
and where $p$ and $r$ are $N$x$N$ matrices and $A^T=(0 \; a)$ and
$a$ are $N$-dimensional vectors. To solve the path integral we need
to integrate this exponent over $\mathrm{d}U=\mathrm{d}\delta r_1
\ldots \mathrm{d}\delta r_N \, \mathrm{d}\delta y_1 \ldots \delta
y_N$. The result is \cite{swanson}
\begin{eqnarray}
\frac{1}{(\mathrm{det}\, M)^{1/2}}\exp\left[-\frac{1}{4}A^T M^{-1} A \right].
\end{eqnarray}
Because $M$ has a zero upper left block, its inverse has a zero
lower right block and, therefore, $A^T M^{-1}A = 0$. Since all the
dependence on the initial and final positions is contained in $A$,
(\ref{superpro2}) is indeed a function only of the initial and final
times. Therefore we may write the superpropagator as
\begin{eqnarray}
J(r(T),y(T),r(0),y(0))=G(T,0)\exp\left\{\frac{i}{\hbar}\tilde{S}[r_e,y_e]
\right\}\exp\left\{-\frac{1}{\hbar}\phi[y_e,y_e]\right\},
\end{eqnarray}
and $G(T,0)$ can be calculated by imposing the normalization of the
reduced density operator. | 3,277 | 28,405 | en |
train | 0.44.10 | \section{Solution of the Equations of Motion.}
Taking the Laplace transform of (\ref{eq22}), we get (with
$F^{''}_a(t)\approx\phi_{xx}(t)/2$)
\begin{eqnarray}
\left[(s^2+\Omega_0^2)-\frac{\gamma^2}{m}\tilde{\phi}_{xx}(s)\right]
\tilde{r}_e(s)=sr(0)+\dot{r}(0).\label{a2-1}
\end{eqnarray}
where $\tilde{f}(s)=\mathcal{L}\{f(t)\}$ is the Laplace transform of
$f(t)$. Using
\begin{eqnarray}
\phi_{xx}(t)=\frac{2}{E_c(0)}\langle
p_x(0)x(t)\rangle_e=A\,e^{-\alpha|t|}\sin{(\omega t)}, \label{a2-3}
\end{eqnarray}
(\ref{a2-1}) becomes
\begin{eqnarray}
\tilde{r}_e(s)=\frac{s[(s+\alpha)^2+\omega^2]r(0)+[(s+\alpha)^2+\omega^2]
\dot{r}(0)}{\{(s^2+\Omega_0^2)[(s+\alpha)^2+\omega^2]-
\frac{\gamma^2}{m}A\omega\}}.\label{a2-4}
\end{eqnarray}
The Heaviside's theorem establishes that if $P(s)$ and $Q(s)$ are
polynomials such that the order of $P(s)$ is smaller than the order
of $Q(s)$, then
\begin{eqnarray}
\mathcal{L}^{-1}\left[\frac{P(s)}{Q(s)}\right]=\sum_{i=1}^{n}
\frac{P(s_i)}{Q'(s_i)} e^{s_it},\label{a2-5}
\end{eqnarray}
where $s_i$ are the roots of $Q(s)=0$ and $Q'(s)$ is the
$s$-derivative of $Q(s)$. Therefore we need the roots of
\begin{eqnarray}
\left[x^2+\left(\frac{\omega_0}{\alpha}\right)^2\right]\left[(x+1)^2+
\left(\frac{\omega}{\alpha}\right)
^2\right]-\frac{\gamma^2}{m}\frac{A\omega}{\alpha^4}=0,
\label{a2-7}
\end{eqnarray}
where $x=s/\alpha$ and $\Omega_0\approx \omega_0$. From Section II
we have
\begin{eqnarray}
\left(\frac{\omega_0}{\alpha}\right)^2\approx 1.6\times 10^{-2}\qquad
\left(\frac{\omega}{\alpha}\right)^2\approx 25 \qquad
\frac{\gamma^2}{m}\frac{A\omega}{\alpha^4}\approx 3\times 10^{-2}.\label{a2-8}
\end{eqnarray}
and the roots of (\ref{a2-7}) are
\begin{eqnarray}
x_1=-1.00-i 5.00 &\qquad& x_2=-1.00+i5.00
\nonumber \\
x_3=-4\times10^{-5}-i0.12 &\qquad& x_4=-4\times10^{-5}+i0.12,\label{a2-9}
\end{eqnarray}
Multiplying these roots by $\alpha$, we get
\begin{eqnarray}
s_1\approx-\alpha-i \omega &\qquad& s_2\approx-\alpha+i\omega
\nonumber \\
s_3\approx-\Lambda-i\omega_0 &\qquad& s_4\approx
-\Lambda+i\omega_0.\label{a2-10}
\end{eqnarray}
The same procedure is applied to (\ref{eq23}). The Laplace transform
of (\ref{eq23}) is written as
\begin{eqnarray}
\tilde{y}_e(s)=\frac{s[(s+\alpha)^2+\omega^2]y(0)+[(s+\alpha)^2+\omega^2]
\dot{y}(0)}{\{(s^2+\Omega_0^2)[(s+\alpha)^2+\omega^2]+
\frac{\gamma^2}{m}A\omega\}}\label{a2-11}
\end{eqnarray}
and the roots are
\begin{eqnarray}
s_1\approx-\alpha-i \omega &\qquad& s_2\approx-\alpha+i\omega
\nonumber \\
s_3\approx \Lambda-i\omega_0 &\qquad& s_4\approx
\Lambda+i\omega_0.\label{a2-12}
\end{eqnarray}
Since we are interested on time scales such that $t\sim 1/\omega_0$,
$s_1$ and $s_2$ are transient solutions and only $s_3$ and $s_4$ are
important. Therefore, turning to the equations (\ref{eq-3}) and
(\ref{eq-4}) and considering times on the scale $t\sim
1/\omega_0$, we see that those equations can be rewritten
approximately as
\begin{eqnarray}
\ddot{r}_e(t)+2\Lambda\dot{r}_e(t)+\Omega_0^2r_e(t)=0,\label{a2-13} \\
\ddot{y}_e(t)-2\Lambda\dot{y}_e(t)+\chi_0^2y_e(t)=0,\label{a2-14}
\end{eqnarray}
where terms proportional to $F(t)$ were disregarded (since they go to
zero for $t\sim 1/\omega_0$) and the convolutions terms were
approximated in the following way
\begin{eqnarray}
\int_0^t\mathrm{d}s\,F(t-s)\dot{r}_e(s)\approx\dot{r}_e(t)
\lim_{t\to\infty}\int_0^{t}\mathrm{d}s
F(t-s).\label{a2-15}
\end{eqnarray}
Thus, $\Lambda$ is given by
\begin{eqnarray}
\Lambda=\frac{\gamma^2}{2m}\lim_{t\to\infty}\int_0^{t}\mathrm{d}s
F(t-s).\label{a2-16}
\end{eqnarray}
Indeed, applying the Laplace transform in (\ref{a2-13}) and
(\ref{a2-14}), we get the roots
\begin{eqnarray}
s_1=-\Lambda-i\omega_0 &\qquad& s_2=-\Lambda+i\omega_0,\label{a2-17}
\end{eqnarray}
for $r_e(t)$ and
\begin{eqnarray}
s_1=\Lambda-i\omega_0 &\qquad& s_2=\Lambda+i\omega_0,\label{a2-18}
\end{eqnarray}
for $y_e(t)$ since $\Omega_0^2,\chi_0^2\gg\Lambda$ and
$\Omega_0^2\approx\chi_0^2\approx\omega_0^2$. Comparing
(\ref{a2-17}) and (\ref{a2-18}) with (\ref{a2-10}) and
(\ref{a2-12}), we conclude that the equations (\ref{a2-13}) and
(\ref{a2-14}) give a good description of the behavior given by
(\ref{eq-3}) and (\ref{eq-4}) for $t\sim 1/\omega_0$.
\end{appendix}
\centerline{\bf Acknowledgements} \noindent This paper was partly
supported by the Brazilian agencies {\bf FAPESP}, under contracts
number 02/04377-7 and 03/12097-7, and {\bf CNPq}. Especial thanks to
S.M.P.
\begin{figure}
\caption{Correlation functions for the NS for
$E_c=0.38$: (a) $\langle p_x (0)x(t)\rangle_e$; (b) $\langle
x(0)x(t)\rangle_e$. The full line shows the numerical results and
the dashed line shows the fitting. The averages were computed using
35000 initial conditions.}
\label{fig2}
\end{figure}
\begin{figure}
\caption{Average oscillator energy at short times
with the NS as chaotic system. $T_0=1/\omega_0$. The dashed line
shows $\langle
E_o(t)\rangle$ and the doted line shows $\langle
E_{or}
\label{fig3}
\end{figure}
\begin{figure}
\caption{Squared width of wave packet
$\sigma^2(T)/\sigma^2$ as given by Eq.(\ref{eq-35}
\label{fig4}
\end{figure}
\begin{figure}
\caption{The full line shows $g(T)$ as given by
Eq.(\ref{eq-49}
\label{fig5}
\end{figure}
\end{document} | 2,286 | 28,405 | en |
train | 0.45.0 | \begin{document}
\title{Stability and convergence analysis of a class of continuous piecewise polynomial approximations for time fractional differential equations}
\section{Introduction}
Fractional calculus, as a generalization of ordinary calculus, has been an intriguing topic for many famous mathematicians since the end of the 17th century. During the last four decades, many scholars have been working on the development of theory for fractional derivatives and integrals, found their way in the world of fractional calculus and their applications. For more detailed information on the historical background, we refer the interested reader to the following books: \cite{Oldhamspanier:1974, Samko:1993, MillerB:1993, Podlubny:1999, Hilfer:2000, Kilbas:2006, Baleanu:2011} and \cite{Herrmann:2014}. Differential equations possessing terms with fractional derivatives in the space- or time- or space-time direction have become very important in many application areas. Particularly, in recent years a huge amount of interesting and surprising fractional models have been proposed. Here, we mention just a few typical applications: in the theory of Hankel transforms \cite{Erdelyi:1940}, in financial models \cite{Scalas:2000, Wyss:2000}, in elasticity theory \cite{Bagleytorvik:1983}, in medical applications \cite{Santamaria:2006, Langlands:2009}, in geology \cite{Benson:2000, Liu:2003}, in physics \cite{Carpinteri:1997,Barkai:2000, Metzler:2000} and many more.
Similar to the work for ordinary differential equations, that has started more than a century earlier, research on numerical methods for time fractional differential equations (tfDEs) started its development. In this paper we consider approximations to tfDEs involving Caputo fractional derivatives of order $0 < \alpha < 1$ in the form of
\begin{equation}
\label{eq:nolinfode}
{^{C}}D^{\alpha}u(t)=f(t,u(t)),\hspace{0.618cm} t\in (0, T]
\end{equation}
with prescribed initial condition $u(0)=u_{0}$. According to \cite{DiethelmK:2004}, it holds that if function $f(t, u(t))$ is continuous and satisfies the Lipschitz condition with respect to the second variable, the problem \eqref{eq:nolinfode} then possesses a unique solution $u(t)\in C([0, T])$. In terms of the numerical approximation of formula \eqref{eq:nolinfode}, we mainly aim at the numerical discretisation to the Caputo fractional derivative, the definition of which we refer to Definition \ref{def:Caputoderiv} in the next section. It is observed that the Caputo fractional derivative of a well-behaved function is an operator combined with the integer-order derivative and the fractional integral, which can be regarded as a convolution of the weakly singular kernel $t^{-\beta} (0<\beta<1)$ and a function. The research on numerical approximations to fractional integral was developed in numerically solving a type of Volterra integral equation
\begin{equation}
\label{eq:nolinfode1}
u(t)=u_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-\xi)^{\alpha-1}f(\xi, u(\xi))\mathrm{d}\xi,
\end{equation}
which is an equivalent form of \eqref{eq:nolinfode} when $f$ satisfies the Lipschitz continuous condition. With respect to numerical approximation for \eqref{eq:nolinfode1}, two general approaches are proposed respectively, i.e, the product integration method and fractional linear multistep methods. In these cases, a general discretized formula to \eqref{eq:nolinfode1} is written as
\begin{equation}
u_{n}=u_{0}+(\Delta t)^{\alpha}\sum_{j=0}^{n}\omega_{n-j}f(t_{j}, u_{j})+(\Delta t)^{\alpha}\sum_{j=0}^{k-1}w_{n,j}f(t_{j}, u_{j}), \hspace{0.5cm} n\ge k.
\end{equation}
The fractional linear multistep methods was originally proposed in \cite{LubichC:1986a} in the mid eighties of the last century. This type of methods is devoted to constructing the power series generated by the convolution quadrature weights $\{\omega_{j}\}_{j=0}^{\infty}$ based on classical implicit linear multistep formulae $(\rho, \sigma)$ with the following relationship
\begin{equation*}
\sum_{j=0}^{\infty}\omega_{j}\xi^{j}=\left(\frac{\sigma(1/\xi)}{\rho(1/\xi)}\right)^{\alpha}.
\end{equation*}
For the motivation behind this idea we refer to \cite{LubichC:1988a}. For this type of methods, the accuracy and stability properties highly benefit from those of the corresponding multistep methods \cite{LubichC:1986b, Lubich:1985}. Another more straightforward approach to generate the weights $\{\omega_{j}\}$ and $\{w_{n,j}\}$ is based on product quadrature method applied to the underlying fractional integral, for examples, to replace the integrand $f(\xi, u(\xi))$ by the piecewise Lagrange interpolation polynomials of degree $k (k\ge 0)$, and approximate the corresponding fractional integral in \eqref{eq:nolinfode1}. On the accuracy and efficiency of this class of methods to a type of nonlinear Volterra integral equation with irregular kernel, we can refer to \cite{Linz:1969, HoogR:1974, CameronS:1984, DixonJ:1985} and recently \cite{Baleanu:2011, LiZeng:2015}.
Other useful approaches such as collocation methods on non-smooth solution are discussed in \cite{Brunner:1985, BrunnerH:1986}. In addition, a series of other approaches were developed such as in \cite{Garappa:2013}, exponential integrators are applied to fractional order problems. Generalized Adams methods and so-called $m$-steps methods are utilized by \cite{Aceto:2014, Aceto:2015}.
In contrast to previous methods, the numerical approach to solve tfDEs in this paper is arrived at through directly approximating fractional derivative combining with numerical differentiation and integration. Recently, a new type of numerical schemes was designed to approximate the Caputo fractional derivative for solving time fractional partial differential equations, such as $L1$ method \cite{LinX:2007}, $L$1-2 method \cite{GaoSZ:2014}, $L$2-$1_\sigma$ method \cite{Alikhanov:2015}. These methods are all based on piecewise linear or quadratic interpolating polynomials approximation. It is natural to generalise the approach by improving the degree of the piecewise polynomial to approximate function that possesses suitable smoothness, in which situation the higher order of accuracy can be obtained. In the next section, we will devote to deriving a series of numerical schemes for formula \eqref{eq:nolinfode} based on constructing piecewise interpolation polynomials on interval $[0, t]$ as the approximations to solution $u(t)$, and consequently, the $\alpha$ order Caputo derivative of the polynomials as the approximation to ${^{C}}D^{\alpha}u(t)$. The local truncation errors of the numerical schemes are discussed correspondingly. The flexibility via choosing different interpolating points on subintervals to construct the piecewise polynomials will produce various schemes under the similar restriction of accuracy order.
In order to study the numerical stability of such methods applying to problem \eqref{eq:nolinfode}, we will examine the behaviour of the numerical method on the linear scalar equation
\begin{equation}
\label{eq:testeqC}
{^{C}}D^{\alpha}u(t)=\lambda u(t), \hspace{0.5cm} \lambda\in\mathbb{C}
\end{equation}
with initial value $u(0)=u_{0}$. It is already shown that the solution of \eqref{eq:testeqC} satisfies that $u(t)\to 0$ as $t\to +\infty$ provided that $|\mathrm{arg}(\lambda)|>\frac{\alpha\pi}{2}~ (-\pi\le \mathrm{arg}(\lambda)\le \pi)$ for arbitrary bounded initial value \cite{LubichC:1986b, MatignonD:1996},
accordingly, it can be studied in seeking those $\lambda$ for which the corresponding numerical solutions preserve the same property as true solution. In fact, several classical numerical stability theories have been constructed on solving problem \eqref{eq:testeqC} in the case of $\alpha=1$ \cite{HairerWN:1991,HairerWN:1987}. Furthermore, there are some efforts on generalising the numerical stability theory on linear multistep methods to integral equations, such as Volterra-type integral equation \cite{LubichC:1983, LubichC:1986b}. It is known that, for example, in the case of all those $\lambda$ satisfying $|\mathrm{arg}(\lambda)|>\frac{\alpha\pi}{2}~(0<\alpha\le 1)$, if the numerical solution has the same asymptotical stability property as true solution, the numerical method is called $A$-stable, and in other case of $\lambda$ lying in the sector with $|\mathrm{arg}(\lambda)|\ge \theta~(\frac{\alpha\pi}{2}<|\theta|\le \pi)$, it is referred to as $A(\theta)$-stable. Inspired by the successive work, we make use of the technique pioneered in \cite{LubichC:1986b}, specialise and refine the results to the fractional case in this paper. We confirm the stability regions and strong stability of the proposed numerical methods, and provide the rigorous analysis on the $A(\frac{\pi}{2})$-stability of some methods. Actually, it can be observed from the numerical experiments that the class of methods possesses the property of $A(\theta)$-stability uniformly for $0<\alpha<1$, and for some specific $\alpha\in (0,1)$, the $A$-stability can be obtained.
The paper is organized as follows. Section \ref{piecewise} introduces the continuous piecewise polynomial approximations of the Caputo derivative. We also derive some useful properties of the weight coefficients and discuss the local truncation errors. Section \ref{stabsection} and Section \ref{conversection} respectively treat the stability and convergence aspects of the numerical schemes when applied to time fractional differential equations. In section \ref{numexpsection} numerical experiments confirm our theoretical considerations with respect to order of convergence and stability restrictions. | 2,614 | 54,748 | en |
train | 0.45.1 | \section{Continuous piecewise polynomial approximation to the Caputo fractional derivative}
\label{piecewise}
We first introduce the fractional derivative in the Caputo sense:
\begin{definition}[\cite{DiethelmK:2004}]\label{def:Caputoderiv}
Let $\alpha>0$, and $n=\lceil \alpha \rceil$, the $\alpha$ order Caputo derivative of function $u(t)$ on $[0,T]$ is defined by
\begin{equation}
{^{C}}D^{\alpha}u(t)=\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}\frac{u^{(n)}(\xi)}{(t-\xi)^{\alpha-n+1}}\mathrm{d}\xi
\end{equation}
whenever $u^{(n)}(t)\in L^{1}[0, T]$. In particular, the Caputo derivative of order $\alpha\in(0,1)$ is defined by
\begin{equation}
\label{eq:Cderiv}
{^C}D^{\alpha} u(t)=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-\xi)^{-\alpha}u^{(1)}(\xi)\mathrm{d}\xi
\end{equation}
whenever $u^{(1)}(t)\in L^{1}[0, T]$.
\end{definition}
In the sequel, we will derive a class of piecewise polynomial approximations to the Caputo fractional derivative of order $\alpha\in(0,1)$. The main idea is as follows.
Let $I=[0, T]$ be an interval, the $M+1$ nodes $\{t_{i}\}_{i=0}^{M}$ define a partition
\begin{equation}\label{par}
0=t_{0}<t_{1}\cdots<t_{M-1}<t_{M}=T.
\end{equation}
If the solution $u(t)$ in \eqref{eq:Cderiv} is assumed to be continuous on the interval $I$, when we think about an piecewise polynomial approximation to $u(t)$, it is reasonable to find the approximate solution at least in the continuous piecewise polynomial space, which is defined by
\begin{equation*}
C_{p}(I)=\{v(t)\in C(I): ~~v(t) \text{ is a polynomial on each subinterval } I_{j}=[t_{j-1}, t_{j}]\}.
\end{equation*}
Specifically, denoting the space of continuous piecewise polynomial of degree at most $k$ by
\begin{equation*}
C_{p}^{k}(I)=\{v(t)\in C(I):~~ v(t)=\sum_{l=0}^{k}a_{j,l}t^{l} \text{ on } I_{j}\},
\end{equation*}
we then construct a class of approximate solutions of $u(t)$ in the space $C_{p}^{k}(I)$. Here the Lagrange interpolation technique by prescribing interpolation conditions on distinct $k+1$ nodes is made use of such that the coefficients $\{a_{j,l}\}$ for $0\le l\le k$ on each $I_{j}$ are uniquely determined. In addition, suppose that $u(t)$ is not a constant function, we only need to focus on the cases of $k\ge 1$ in view of the continuity restriction. The choice of the interpolating points is provided in the following way.
We define a class of polynomials $p_{j, q}^{k}(t)$ which are of degree $k\ge 1$ and have a compact support $I_{j}$. The coefficients of the polynomials are uniquely determined by the following $k+1$ interpolation conditions
\begin{equation}\label{eq:interpcond}
p_{j, q}^{k}(t_{n})=u(t_{n}),\hspace{0.618cm} n=j+q-1, j+q-2,\cdots, j+q-k-1.
\end{equation}
Here the index $q$ records the number of shifts of the $k+1$ interpolating nodes $\{t_{n}\}_{n=j-1-k}^{j-1}$, and the sign of $q$ indicates the direction of the shift. Based on \eqref{eq:interpcond}, the polynomials can be represented by a Lagrange form
\begin{equation}\label{eq:pjqk}
p_{j, q}^{k}(t)=\sum_{n=j+q-k-1}^{j+q-1}\prod_{m=j+q-k-1 \atop m\ne n}^{j+q-1}\frac{t-t_{m}}{t_{n}-t_{m}} u(t_{n}).
\end{equation}
In particular, if the partition \eqref{par} is equidistant, i.e., $t_{n}=n\Delta t$ and $\Delta t=\frac{T}{M}$ as $M\in \mathbb{N}^{+}$, the alternative Newton expression is given by
\begin{equation}\label{eq:piqk}
p_{j, q}^{k}(t)=\sum_{n=0}^{k}\frac{\nabla^{n}u(t_{j+q-1})}{n!(\Delta t)^{n}}\prod_{l=0}^{n-1}(t-t_{j+q-1-l}).
\end{equation}
For the convenience of notation, we then rewrite \eqref{eq:piqk} by changing the variable $t=t_{j-1}+s\Delta t$ to obtain
\begin{equation}
\label{eq:pk}
p_{j,q}^{k}(t)=p_{j,q}^{k}(t_{j-1}+s\Delta t)=\sum_{r=0}^{k}\binom{s-q+r-1}{r}\nabla^{r}u(t_{j+q-1}),
\end{equation}
where the $r$-th order backward difference operator $\nabla^{r}$ is commonly defined by
\begin{equation*}
\nabla^{0} u(t_{i})=u(t_{i}),\hspace{0.4cm} \nabla^{r}u(t_{i})=\nabla^{r-1}u(t_{i})-\nabla^{r-1}u(t_{i-1})
\end{equation*}
and $\binom{s-q+r-1}{r}$ is the binomial coefficient.
In the following, we construct a class of approximate solution $P_{i}^{k}(t)\in C_{p}^{k}(I)$ to $u(t)$ on the uniform grid for $1\le i\le k\le 6$. The general representations are proposed by
\begin{equation}\label{eq:Pik}
P_{i}^{k}(t)=\sum_{j=1}^{k-i}p_{j,k-j}^{k-1}(t)+\sum_{j=k}^{n}p_{j-i+1,i}^{k}(t)+\sum_{j=n-i+2}^{n}p_{j,n+1-j}^{k}(t)
\end{equation}
for $t\in (t_{n-1}, t_{n}]$ and $1\le n\le M$, where $\sum_{j=1}^{k-i}p_{j,k-j}^{k-1}(t)=0$ and $\sum_{j=n-i+2}^{n}p_{j,n+1-j}^{k}(t)=0$ if $k-i<1$ and $n-i+2>n$, respectively.
\begin{remark}\label{re:1}
The construction of polynomials $P_{i}^{k}(t)$ is mainly based on the continuity requirement on interval $I$, i.e., the interpolation conditions
\begin{equation}
\label{eq:2.4}
p_{j,q}^{k}(t_{n})=u(t_{n}),\hspace{0.618cm} n=j-1,\hspace{0.1cm} j
\end{equation}
should be satisfied. It yields that on each subinterval $I_{j}$, according to \eqref{eq:interpcond}, both conditions $j+q-1\ge j$ and $j+q-k-1\le j-1$ should be satisfied, which indicates $1\le q\le k$. Therefore, in the case of $k=1$, there is a unique continuous piecewise linear polynomial, denoted by $P_{1}^{1}(t)$, in the space $C_{p}^{1}(I)$. According to \eqref{eq:Pik}, it is expressed by
\begin{equation*}\label{eq:P11}
P_{1}^{1}(t)=\sum_{j=1}^{n}p_{j,1}^{1}(t),
\end{equation*}
In the other case of $k=2$, there are three options on each $I_{j}$, that is, $p_{j,1}^{1}(t)$, $p_{j,1}^{2}(t)$ and $p_{j,2}^{2}(t)$ to constitute the interpolating polynomial that belongs to space $C_{p}^{2}(I)$. It is known that the construction of $P^{2}(t)$ is therefore not unique. In order to preserve the convolution property as much as possible, here we propose three available continuous piecewise polynomials in the forms of
\begin{equation}\label{eq:P2}
P_{1}^{2}(t)=p_{1,1}^{1}(t)+\sum_{j=2}^{n}p_{j,1}^{2}(t),\hspace{0.618cm} P_{2}^{2}(t)=\sum_{j=1}^{n-1}p_{j,2}^{2}(t)+p_{n,1}^{2}(t)
\end{equation}
and
\begin{equation*}\label{eq:P23}
P_{3}^{2}(t)=p_{1,2}^{2}(t)+\sum_{j=2}^{n}p_{j,1}^{2}(t)
\end{equation*}
when $t\in (t_{n-1}, t_{n}]$. In addition, as shown in \eqref{eq:Pik}, we restrict our further discussion to the case $i\le k$, and it is because in that situation, the corresponding discretized operators $D_{k,i}^{\alpha}u_{n}$ defined in \eqref{Dki1} can be computed recursively when the least starting values are prescribed.
\end{remark}
As a consequence, the operator
\begin{equation}
\label{Dki}
D_{k,i}^{\alpha}u(t)=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-\xi)^{-\alpha}\frac{\mathrm{d} P_{i}^{k}}{\mathrm{d} \xi}\mathrm{d}\xi
\end{equation}
is proposed on $t\in I$ as the approximations to ${^{C}}D^{\alpha}u(t)$. In the case of $t=t_{n}$, formula \eqref{Dki} can also be rewritten as
\begin{equation}
\label{Dki1}
\begin{split}
D_{k,i}^{\alpha}u_{n}&=\frac{(\Delta t)^{-\alpha}}{\Gamma(1-\alpha)}\sum_{j=1}^{n}\int_{0}^{1}(n-j+1-s)^{-\alpha}\frac{\mathrm{d} P_{i}^{k}(t_{j-1}+s\Delta t)}{\mathrm{d}s}\mathrm{d}s \\
&=(\Delta t)^{-\alpha}\sum_{j=0}^{k-1}w_{n,j}^{(k,i)}u_{j}+(\Delta t)^{-\alpha}\sum_{j=0}^{n}\omega_{n-j}^{(k,i)}u_{j},
\end{split}
\end{equation}
where $u_{n}:=u(t_{n})$. | 2,788 | 54,748 | en |
train | 0.45.2 | In the following part, we will present the explicit representations of the weight coefficients $\{w_{n,j}^{(k,i)}\}$ and $\{\omega_{j}^{(k,i)}\}$ when $1\le i\le k\le 3$ as examples. First, define that
\begin{equation}
\label{ljqr}
I_{n,q}^{r}=\left\{
\begin{array}{cl}
\frac{1}{\Gamma(1-\alpha)}\int\limits_{0}^{1}(n+1-s)^{-\alpha}\mathrm{d}\binom{s-q+r-1}{r},& n\ge 0, \\
0, & n<0, \\
\end{array}
\right.
\end{equation}
where $q, r\in\mathbb{N}^{+}$ and $n\in\mathbb{Z}$. In addition, note that
\begin{equation*}
\begin{split}
& I_{n}:=I_{n,q}^{1},\hspace{0.3cm} \forall~q=1,2,\cdots, \\
&\nabla^{k}I_{n,q}^{r}=\nabla^{k-1}I_{n,q}^{r}-\nabla^{k-1}I_{n-1,q}^{r}, \hspace{0.3cm} \forall~k\in\mathbb{N}^{+}.
\end{split}
\end{equation*}
Then the weight coefficients can be expressed in terms of integrals $I_{n,q}^{r}$ by
\begin{equation*}
\left\{
\begin{split}
(k,i)=(1,1): &\hspace{0.2cm}w_{m,0}=-I_{m},\hspace{0.3cm} m\ge 1, \hspace{0.4cm} \omega_{n}=\nabla I_{n},\hspace{0.3cm}n\ge 0, \\
(k,i)=(2,1): &\hspace{0.2cm}w_{m,0}=2I_{m-1,1}^{2}-I_{m,1}^{2}-I_{m}, \hspace{0.4cm} w_{m,1}=-I_{m-1,1}^{2},\hspace{0.2cm}m\ge 2, \\
&\hspace{0.2cm}\omega_{n}=\nabla I_{n}+\nabla^{2} I_{n,1}^{2},\hspace{0.3cm} n\ge 0, \\
(k,i)=(2,2): &\hspace{0.2cm}w_{m,0}=-\nabla I_{m+1,1}^{2}+I_{m,2}^{2}, \hspace{0.4cm}w_{m,1}=-I_{m,1}^{2},\hspace{0.3cm} m\ge 2, \\
&\hspace{0.2cm}\omega_{0}=I_{0}+I_{1}+I_{0,1}^{2}+I_{1,2}^{2},\hspace{0.4cm}\omega_{1}=\nabla I_{2}-I_{0}+I_{2,2}^{2}-2I_{0,1}^{2}-2I_{1,2}^{2}, \\
&\hspace{0.2cm}\omega_{2}=\nabla I_{3}+\nabla^{2}I_{3,2}^{2}+I_{0,1}^{2}, \hspace{0.4cm} \omega_{n}=\nabla I_{n+1}+\nabla^{2} I_{n+1,2}^{2},\hspace{0.2cm}n\ge 3, \\
\end{split}
\right.
\end{equation*}
and by more complicated formulae of forms
\begin{description}
\item [i).] $(k,i)=(3,1)$,
\begin{equation}\label{eq:ki31}
\left\{
\begin{split}
&w_{m,0}=-\nabla I_{m}-I_{m,1}^{2}+2I_{m-1,1}^{2}+I_{m-1,2}^{2}-I_{m,1}^{3}+3I_{m-1,1}^{3}-3I_{m-2,1}^{3},\\
&w_{m,1}=-2I_{m-1}-2I_{m-1,2}^{2}-I_{m-1,1}^{2}-I_{m-1,1}^{3}+3I_{m-2,1}^{3},\\
&w_{m,2}=I_{m-1}+I_{m-1,2}^{2}-I_{m-2,1}^{3},\hspace{0.5cm} m\ge 3, \\
&\omega_{n}=\nabla I_{n}+\nabla^{2}I_{n,1}^{2}+\nabla^{3}I_{n,1}^{3}, \hspace{0.5cm} n\ge 0, \\
\end{split}
\right.
\end{equation}
\item [ii).] $(k,i)=(3,2)$,
\begin{equation}\label{eq:ki32}
\left\{
\begin{split}
&w_{m,0}=-\nabla I_{m+1}-I_{m+1,2}^{2}+2I_{m,2}^{2}-I_{m+1,2}^{3}+3I_{m,2}^{3}-3I_{m-1,2}^{3},\\
&w_{m,1}=-I_{m}-I_{m,2}^{2}-I_{m,2}^{3}+3I_{m-1,2}^{3},\\
&w_{m,2}=-I_{m-1,2}^{3},\hspace{0.3cm}m\ge 3, \\
&\omega_{0}=I_{0}+I_{1}+I_{1,2}^{2}+I_{0,1}^{2}+I_{1,2}^{3}+I_{0,1}^{3}, \\
&\omega_{1}=\nabla I_{2}-I_{0}+I_{2,2}^{2}-2I_{1,2}^{2}-2I_{0,1}^{2}+I_{2,2}^{3}-3I_{1,2}^{3}-3I_{0,1}^{3}, \\
&\omega_{2}=\nabla I_{3}+\nabla^{2}I_{3,2}^{2}+I_{0,1}^{2}+I_{3,2}^{3}-3I_{2,2}^{3}+3I_{1,2}^{3}+3I_{0,1}^{3}, \\
&\omega_{3}=\nabla I_{4}+\nabla^{2}I_{4,2}^{2}+\nabla^{3}I_{4,2}^{3}-I_{0,1}^{3}, \\
&\omega_{n}=\nabla I_{n+1}+\nabla^{2}I_{n+1,2}^{2}+\nabla^{3}I_{n+1,2}^{3}, \hspace{0.3cm} n\ge 4,\\
\end{split}
\right.
\end{equation}
\item [iii).]$(k,i)=(3,3)$,
\begin{equation}\label{eq:ki33}
\left\{
\begin{split}
&w_{m,0}=-\nabla I_{m+2}-\nabla^{2} I_{m+2,3}^{2}-I_{m+2,3}^{3}+3I_{m+1,3}^{3}-3I_{m,3}^{3},\\
&w_{m,1}=-\nabla I_{m+1}-I_{m+1,3}^{2}+2I_{m,3}^{2}-I_{m+1,3}^{3}+3I_{m,3}^{3},\\
&w_{m,2}=-I_{m}-I_{m,3}^{2}-I_{m,3}^{3},\hspace{0.3cm}m\ge 3, \\
&\omega_{0}=I_{0}+I_{1}+I_{2}+I_{0,1}^{2}+I_{1,2}^{2}+I_{2,3}^{2}+I_{0,1}^{3}+I_{1,2}^{3}+I_{2,3}^{3}, \\
&\omega_{1}=\nabla I_{3}-I_{0}-I_{1}+I_{3,3}^{2}-2I_{2,3}^{2}-2I_{1,2}^{2}-2I_{0,1}^{2}+I_{3,3}^{3}-3I_{2,3}^{3}-3I_{1,2}^{3}-3I_{0,1}^{3},\\
&\omega_{2}=\nabla I_{4}+\nabla^{2}I_{4,3}^{2}+I_{1,2}^{2}+I_{0,1}^{2}+I_{4,3}^{3}-3I_{3,3}^{3}+3I_{2,3}^{3}+3I_{0,1}^{3}+3I_{1,2}^{3}, \\
&\omega_{3}=\nabla I_{5}+\nabla^{2} I_{5,3}^{2}+\nabla^{3} I_{5,3}^{3}-I_{1,2}^{3}-I_{0,1}^{3},\\
&\omega_{n}=\nabla I_{n+2}+\nabla^{2}I_{n+2,3}^{2}+\nabla^{3}I_{n+2,3}^{3}, \hspace{0.3cm}n\ge 4.\\
\end{split}
\right.
\end{equation}
\end{description}
In addition, it is observed that when $\alpha\to1$, the difference operator $D_{k,i}^{\alpha}u_{n}$ in $\eqref{Dki1}$ recovers to the $k$-step BDF method.
\begin{remark}
The construction process of operator \eqref{Dki} can be extended to the case of $\alpha>1$ as well. In a general case of $\lceil \alpha \rceil-1<\alpha<\lceil \alpha \rceil$, the interpolating polynomials $P_{i}^{k}(t)\in C_{p}^{k}(I)$ could be constructed as the approximations to $u(t)$ under the condition that $k\ge \lceil \alpha \rceil$, and the $\alpha$ order Caputo derivative of $P_{i}^{k}(t)$ are proposed in an analogous way by
\begin{equation*}
D_{k,i}^{\alpha}u(t)=\frac{1}{\Gamma(\lceil \alpha \rceil-\alpha)}\int_{0}^{t}(t-\xi)^{-\alpha+\lceil \alpha \rceil-1}\frac{\mathrm{d^{\lceil \alpha \rceil}} P_{i}^{k}}{\mathrm{d} \xi^{\lceil \alpha \rceil}}\mathrm{d}\xi
\end{equation*}
as the approximation to ${^{C}}D^{\alpha}u(t)$. Here the condition of $k\ge \lceil \alpha \rceil$ is required such that the $\lceil \alpha \rceil$ order derivative of $P_{i}^{k}(t)$ is nonzero a.e.. We take the case of $\lceil \alpha \rceil=2$ as an example. Assume that $\beta=\alpha-1\in (0,1)$, the polynomials $P_{i}^{2}(t)$ denoted by \eqref{eq:P2} are in the space $C_{p}^{2}(I)$, and it follows
\begin{equation*}
\begin{split}
D_{2,i}^{\alpha}u(t_{n})=&\frac{1}{\Gamma(1-\beta)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_{j}}(t_{n}-\xi)^{-\beta}\frac{\mathrm{d^2} P_{i}^{2}}{\mathrm{d} \xi^2}\mathrm{d}\xi \\
=&\frac{(\Delta t)^{-\alpha}}{\Gamma(1-\beta)}\sum_{j=1}^{n}\int_{0}^{1}(n-j+1-s)^{-\beta}\frac{\mathrm{d}^{2} P_{i}^{2}(t_{j-1}+s\Delta t)}{\mathrm{d}s^{2}}\mathrm{d}s,
\end{split}
\end{equation*}
where the last equality holds based on the relation $\frac{\mathrm{d}^{2} P_{i}^{k}(\xi(s))}{\mathrm{d}s^{2}}=\frac{\mathrm{d^2} P_{i}^{k}(\xi)}{\mathrm{d} \xi^2}\left(\frac{\mathrm{d}\xi}{\mathrm{d}s}\right)^{2}+\frac{\mathrm{d} P_{i}^{k}(\xi)}{\mathrm{d}\xi}\frac{\mathrm{d}^{2}\xi}{\mathrm{d}s^{2}}$.
Moreover, it can be rewritten as a form analogous to \eqref{Dki1}, and the corresponding weights coefficients $\{w_{n,i}\}$ and $\{\omega_{n}\}$ are therefore derived by
\begin{equation}\left\{
\begin{split}
&\omega_{j}^{(2,1)}=\nabla^{2} I_{j},\hspace{0.4cm} j\ge 0,\hspace{0.4cm} w_{n,0}^{(2,1)}=-\left(I_{n}-2I_{n-1}\right), \hspace{0.4cm}w_{n,1}^{(2,1)}=-I_{n-1},\hspace{0.4cm}n\ge 2, \\
&\omega_{0}^{(2,2)}=I_{0}+I_{1},\hspace{0.5cm}\omega_{1}^{(2,2)}=I_{2}-2I_{1}-2I_{0}, \hspace{0.5cm} \omega_{2}^{(2,2)}=\nabla^{2}I_{3}+I_{0}, \\
&\omega_{j}^{(2,2)}=\nabla^{2}I_{j+1}, \hspace{0.4cm}j\ge 3, \hspace{0.5cm} w_{n,0}^{(2,2)}=-I_{n+1}+2I_{n}, \hspace{0.4cm} w_{n,1}^{(2,2)}=-I_{n},\hspace{0.4cm}n\ge 2.
\end{split}
\right.
\end{equation}
Here the integrals $I_{n,q}^{r}=I_{n,q}^{r}(\beta)$ are defined by \eqref{ljqr} where the index $\alpha$ is replaced by $\beta$.
\end{remark} | 3,391 | 54,748 | en |
train | 0.45.3 | \subsection{Complete monotonicity and error analysis}
First, we explore the completely monotonic property of the sequence $\{I_{n,q}^{r}\}_{n=0}^{\infty}$.
\begin{lemma}
\label{le:Ijqr}
Assume that $I_{n,q}^{r}$ is defined by \eqref{ljqr}, then for $n\ge k$ with $k\in \mathbb{N}$, it holds that
\begin{equation}
\label{eq:r<q}
(-1)^{k+r+1}\nabla^{k} I_{n,q}^{r}\ge 0
\end{equation}
in the case of $r\le q$, and
\begin{equation}
\label{eq:r>q}
(-1)^{k+q+1}\nabla^{k} I_{n,q}^{r}\ge 0
\end{equation}
in the case of $ r>q$.
\end{lemma}
\begin{proof}
We begin with the case of $r\le q$, according to the definition of $I_{n,q}^{r}$ in \eqref{ljqr}, it holds that
\begin{equation}\label{eq:inqr}
\begin{split}
I_{n,q}^{r}=&\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\mathrm{d}\binom{s-q+r-1}{r} \\
=&\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\sum_{n=0}^{r-1}\frac{1}{(s-q+n)}\binom{s-q+r-1}{r}\mathrm{d}s,
\end{split}
\end{equation}
since $(s-q+n)\le 0$ for $0\le s\le 1$ and $n=0,\cdots,r-1$, it yields that $(-1)^{r}\binom{s-q+r-1}{r}\ge 0$,
and consequently $(-1)^{r+1}\frac{\mathrm{d}}{\mathrm{d}s}\binom{s-q+r-1}{r}\ge 0$, combined with $(n+1-s)^{-\alpha}>0$ for any $n\ge 0$ and $\alpha>0$, it leads to $(-1)^{r+1}I_{n,q}^{r}\ge 0$. In addition, by definition, we may see that
\begin{equation}\label{eq:nablainqr}
\begin{split}
\nabla I_{n,q}^{r}=&\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}\Big((n+1-s)^{-\alpha}-(n-s)^{-\alpha}\Big)\mathrm{d}\binom{s-q+r-1}{r} \\
=&\frac{-\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}\int_{n}^{n+1}(\xi-s)^{-\alpha-1}\mathrm{d}\xi\mathrm{d}\binom{s-q+r-1}{r} \\
=&\frac{-\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}\int_{0}^{1}(\xi+n-s)^{-\alpha-1}\mathrm{d}\xi\mathrm{d}\binom{s-q+r-1}{r},
\end{split}
\end{equation}
with $(\xi+n-s)^{-\alpha-1}\ge 0$ for $n\ge 1$ and $0\le \xi, s\le 1$, then $(-1)^{r+2}\nabla I_{n,q}^{r}\ge 0$.
Assume that for $k\ge 2$, it holds
\begin{equation*}
\nabla^{k-1}I_{n,q}^{r}=\frac{(-\alpha)_{k-1}}{\Gamma(1-\alpha)}\int_{[0,1]^{k}}(\sum_{i=1}^{k-1}\xi_{i}+n-k+2-s)^{-\alpha-k+1}\mathrm{d}^{k-1}\mathbf{\xi}\mathrm{d}\binom{s-q+r-1}{r},
\end{equation*}
where denote that $(\alpha)_{k-1}=\alpha(\alpha-1)\cdots(\alpha-k+2)$ and $\mathrm{d}^{k-1}\mathbf{\xi}=\mathrm{d}\xi_{1}\cdots\mathrm{d}\xi_{k-1}$, then
\begin{equation}\label{eq:nablakinqr}
\begin{split}
\nabla^{k}I_{n,q}^{r}&=\nabla^{k-1}I_{n,q}^{r}-\nabla^{k-1}I_{n-1,q}^{r} \\
&=\frac{(-\alpha)_{k-1}}{\Gamma(1-\alpha)}\int_{[0,1]^{k}}\nabla(\sum_{i=1}^{k-1}\xi_{i}+n-k+2-s)^{-\alpha-k+1}\mathrm{d}^{k-1}\mathbf{\xi}\mathrm{d}\binom{s-q+r-1}{r} \\
&=\frac{(-\alpha)_{k}}{\Gamma(1-\alpha)}\int_{[0,1]^{k}}\int_{n+1}^{n+2}(\sum_{i=1}^{k}\xi_{i}-k-s)^{-\alpha-k}\mathrm{d}\xi_{k}\mathrm{d}^{k-1}\mathbf{\xi}\mathrm{d}\binom{s-q+r-1}{r} \\
&=\frac{(-\alpha)_{k}}{\Gamma(1-\alpha)}\int_{[0,1]^{k+1}}(\sum_{i=1}^{k}\xi_{i}+n-k+1-s)^{-\alpha-k}\mathrm{d}^{k}\mathbf{\xi}\mathrm{d}\binom{s-q+r-1}{r}.
\end{split}
\end{equation}
Since $(\sum\limits_{i=1}^{k}\xi_{i}+n-k+1-s)\ge 0$ for $n\ge k\ge 1$ and $0\le\xi_{i}, s\le 1$, then \eqref{eq:r<q} holds.
In the other case of $r\ge q+1$, integrating by part yields that
\begin{equation}\label{eq:inqr1}
\begin{split}
I_{n,q}^{r}&=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\mathrm{d}\binom{s-q+r-1}{r} \\
&=\frac{-\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha-1}\binom{s-q+r-1}{r}\mathrm{d}s,
\end{split}
\end{equation}
since $\binom{s-q+r-1}{r}$ includes a factor $s(s-1)$ for $r\ge q+1, q\in\mathbb{N}^{+}$. The sign of $\binom{s-q+r-1}{r} $ is the same with that of $\prod\limits_{i=1}^{q}(s-i)$, thus $(-1)^{q}\binom{s-q+r-1}{r}\ge 0$, and it holds that $(-1)^{q+1}I_{n,q}^{r}\ge 0$ for $n\ge 0$.
Furthermore, the induction process demonstrates that
\begin{equation}\label{eq:nablakinqr1}
\nabla^{k}I_{n,q}^{r}=\frac{(-\alpha)_{k+1}}{\Gamma(1-\alpha)}\int_{[0, 1]^{k+1}}(\sum_{i=1}^{k}\xi_{i}+n-k+1-s)^{-\alpha-k-1}\binom{s-q+r-1}{r}\mathrm{d}^{k}\mathbf{\xi}\mathrm{d}s
\end{equation}
for $n\ge k\ge 1$, which arrives at \eqref{eq:r>q}.
\end{proof}
Moreover, we discuss the complete monotonicity of a general class of sequences.
\begin{lemma}
\label{le:1.5}
The sequence $\{s_{n}\}_{n=0}^{\infty}$ is defined by
\begin{equation*}
s_{n}=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\varphi(s)\mathrm{d}s, \hspace{0.5cm} n\ge 0,
\end{equation*}
where $\varphi(s)\ge 0$ for $0\le s\le 1$. Then for $n\ge k$, it holds that $(-1)^{k}\nabla^{k}s_{n}\ge 0$.
\end{lemma}
\begin{proof}
It is easy to check that $s_{n}\ge 0$ for all $n\ge 0$, since for $n\ge 0$, $0\le s\le 1$, it holds that $(n+1-s)^{-\alpha}>0$ and $\varphi(s)\ge0$ by assumption. The definition of $s_{n}$ implies that
\begin{equation*}
\begin{split}
\nabla s_{n}&=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}\left((n+1-s)^{-\alpha}-(n-s)^{-\alpha}\right)\varphi(s)\mathrm{d}s \\
&=\frac{-\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}\int_{0}^{1}(n+\xi-s)^{-\alpha-1}\varphi(s)\mathrm{d}\xi\mathrm{d}s, \\
\end{split}
\end{equation*}
since $(n+\xi-s)^{-\alpha-1}>0$ and $\varphi(s)\ge 0$ for $n\ge 1$ and $0\le s, \xi\le 1$, thus $\nabla s_{n}\le 0$ holds.
Therefore an induction process yields that
\begin{equation*}
\nabla^{k} s_{n}=\frac{(-\alpha)_{n}}{\Gamma(1-\alpha)}\int_{[0, 1]^{k+1}}(\sum_{i=1}^{k}\xi_{i}+n-k+1-s)^{-\alpha-k}\varphi(s)\mathrm{d}^{k}\xi\mathrm{d}s.
\end{equation*}
Since for $n\ge k$ and $0\le s,\xi_{i}\le 1$, it holds that $\sum\limits_{i=1}^{k}\xi_{i}+n-k+1-s)^{-\alpha-k}\varphi(s)\ge 0$, thus we can obtain that $(-1)^{k}\nabla^{k}s_{n}\ge 0$ for $n\ge k$.
\end{proof} | 2,623 | 54,748 | en |
train | 0.45.4 | Moreover, we discuss the complete monotonicity of a general class of sequences.
\begin{lemma}
\label{le:1.5}
The sequence $\{s_{n}\}_{n=0}^{\infty}$ is defined by
\begin{equation*}
s_{n}=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\varphi(s)\mathrm{d}s, \hspace{0.5cm} n\ge 0,
\end{equation*}
where $\varphi(s)\ge 0$ for $0\le s\le 1$. Then for $n\ge k$, it holds that $(-1)^{k}\nabla^{k}s_{n}\ge 0$.
\end{lemma}
\begin{proof}
It is easy to check that $s_{n}\ge 0$ for all $n\ge 0$, since for $n\ge 0$, $0\le s\le 1$, it holds that $(n+1-s)^{-\alpha}>0$ and $\varphi(s)\ge0$ by assumption. The definition of $s_{n}$ implies that
\begin{equation*}
\begin{split}
\nabla s_{n}&=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}\left((n+1-s)^{-\alpha}-(n-s)^{-\alpha}\right)\varphi(s)\mathrm{d}s \\
&=\frac{-\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}\int_{0}^{1}(n+\xi-s)^{-\alpha-1}\varphi(s)\mathrm{d}\xi\mathrm{d}s, \\
\end{split}
\end{equation*}
since $(n+\xi-s)^{-\alpha-1}>0$ and $\varphi(s)\ge 0$ for $n\ge 1$ and $0\le s, \xi\le 1$, thus $\nabla s_{n}\le 0$ holds.
Therefore an induction process yields that
\begin{equation*}
\nabla^{k} s_{n}=\frac{(-\alpha)_{n}}{\Gamma(1-\alpha)}\int_{[0, 1]^{k+1}}(\sum_{i=1}^{k}\xi_{i}+n-k+1-s)^{-\alpha-k}\varphi(s)\mathrm{d}^{k}\xi\mathrm{d}s.
\end{equation*}
Since for $n\ge k$ and $0\le s,\xi_{i}\le 1$, it holds that $\sum\limits_{i=1}^{k}\xi_{i}+n-k+1-s)^{-\alpha-k}\varphi(s)\ge 0$, thus we can obtain that $(-1)^{k}\nabla^{k}s_{n}\ge 0$ for $n\ge k$.
\end{proof}
Next, we construct the numerical scheme
\begin{equation}
\label{eq:nnonliode}
D_{k,i}^{\alpha} u_{n}=f(t_{n},u_{n}), \hspace{0.5cm} n\ge k,
\end{equation}
as the approximation to problem \eqref{eq:nolinfode} with prescribed starting values, and define the local truncation error of the $n$-th step by
\begin{equation}
\label{eq:trunc}
\tau_{n}^{(k,i)}=D_{k,i}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n}), \hspace{0.5cm} n\ge k,
\hspace{0.2cm}n\in\mathbb{N}^{+},
\end{equation}
where $u(t)$ is the exact solution of problem \eqref{eq:nolinfode}.
\begin{theorem}
\label{th:errorestmat}
Assume that $u(t)\in C^{k+1}[0, T]$ and $0<\alpha<1$, it holds that
\begin{equation}\label{eq:truncki}
D_{k,i}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})=O\left((t_{n-k+i})^{-\alpha-1}\Delta t^{k+1}+\Delta t^{k+1-\alpha}\right),\hspace{0.4cm}
\end{equation}
for $n\ge k$ in the cases of $1\le i<k\le 6$. In particular
\begin{equation}\label{eq:trunckk}
D_{k,k}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})=O(\Delta t^{k+1-\alpha}), \hspace{0.4cm} k=1,\cdots,6
\end{equation}
holds uniformly for $n\ge k$.
\end{theorem}
\begin{proof}
According to \eqref{eq:pk}, it holds that
\begin{equation}
\label{eq:2.9}
p_{j,q}^{k}(t)-u(t)=u^{(k+1)}(\xi_{j})\binom{s-q+k}{k+1}(\Delta t)^{k+1},
\end{equation}
where $t=t_{j-1}+s\Delta t$ with $0\le s\le 1$ and $t_{j+q-k-1}\le \xi_{j}\le t_{j+q-1}$.
Inspired by \cite{GaoSZ:2014}, making use of the integration by part technique, we arrive at
\begin{equation}\label{eq:2.23}
\begin{split}
D_{k,i}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})&=\frac{1}{\Gamma(1-\alpha)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_{j}}(t_{n}-t)^{-\alpha}
\left(\frac{\mathrm{d} P_{i}^{k}(t)}{\mathrm{d}t}-\frac{\mathrm{d} u(t)}{\mathrm{d} t}\right)\mathrm{d}t \\
&=\frac{-\alpha}{\Gamma(1-\alpha)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_{j}}(t_{n}-t)^{-\alpha-1}\left(P_{i}^{k}(t)-u(t)\right)\mathrm{d}t \\
&=\frac{-\alpha (\Delta t)^{-\alpha}}{\Gamma(1-\alpha)}\sum_{j=1}^{n}\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\left(P_{i}^{k}(t_{j-1}+s\Delta t)-u(t_{j-1}+s\Delta t)\right)\mathrm{d}s
\end{split}
\end{equation}
for $n\ge k$, which is based on the conditions of \eqref{eq:2.4} and \eqref{eq:2.9}.
From the general representation of $P_{i}^{k}(t)$ in \eqref{eq:Pik}, it is known that for $k-i\ge 1$, the polynomials of degree $(k-1)$ are used on subinterval $\cup_{j=1}^{k-i}I_{j}$ to construct $P_{i}^{k}(t)$, in the other case of $k=i$, the polynomials of degree $k$ are chosen on each subinterval $I_{j}$ instead. Therefore, we next consider the two cases seperately.
Substituting \eqref{eq:Pik} and \eqref{eq:2.9} into the last equivalent formula of \eqref{eq:2.23}, if $k=i$, one obtains
\begin{equation*}
\begin{split}
|D_{k,k}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})|&\le\frac{\alpha\left(\Delta t\right)^{k+1-\alpha} }{\Gamma(1-\alpha)}\max_{\xi\in I}|u^{(k+1)}(\xi)|\Big(\sum_{j=1}^{n-k+1}\Big|\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\binom{s}{k+1}\mathrm{d}s\Big| \\
&+\sum_{j=n-k+2}^{n}\Big|\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\binom{s+k-n-1+j}{k+1}\mathrm{d}s\Big|\Big),
\end{split}
\end{equation*}
and if $1\le i\le k-1$, one has
\begin{equation*}
\begin{split}
|D_{k,i}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})|&\le \frac{\alpha\left(\Delta t\right)^{-\alpha} }{\Gamma(1-\alpha)}\Big((\Delta t)^{k}\max_{\xi\in \cup_{j=1}^{k-i}I_{j}}|u^{(k)}(\xi)|\sum_{j=1}^{k-i}\Big|\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\binom{s+j-1}{k}\mathrm{d}s\Big| \\
&+(\Delta t)^{k+1}\max_{\xi\in I}|u^{(k+1)}(\xi)|\sum_{j=k-i+1}^{n-i+1}\Big|\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\binom{s+k-i}{k+1}\mathrm{d}s\Big| \\
&+(\Delta t)^{k+1}\max_{\xi\in I}|u^{(k+1)}(\xi)|\sum_{j=n-i+2}^{n}\Big|\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\binom{s+k-n-i+j}{k+1}\mathrm{d}s\Big|\Big).
\end{split}
\end{equation*}
Since for any $q\le k$ and $q, k\in \mathbb{N}^{+}$, the factor $(1-s)$ is included in $\binom{s-q+k}{k+1}$ and $\frac{1}{1-s}\binom{s-q+k}{k+1}$ is bounded for $0\le s
\le 1$, thus we can obtain that
\begin{equation*}
\begin{split}
|D_{k,k}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})|\le&\frac{\alpha\left(\Delta t\right)^{k+1-\alpha} }{\Gamma(1-\alpha)}C^{(k)}\sum_{j=1}^{n}\int_{0}^{1}(n-j+1-s)^{-\alpha-1}(1-s)\mathrm{d}s \\
\le&\frac{\alpha\left(\Delta t\right)^{k+1-\alpha} }{\Gamma(1-\alpha)}C^{(k)}\Big(\sum_{j=1}^{n-1}\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\mathrm{d}s+\int_{0}^{1}(1-s)^{-\alpha}\mathrm{d}s\Big) \\
\le&\left(\Delta t\right)^{k+1-\alpha}C^{(k)}
\Big(\frac{1}{\Gamma(1-\alpha)}+\frac{1}{\Gamma(2-\alpha)}\Big),
\end{split}
\end{equation*}
where $C^{(k)}$ is bounded relevant to $u^{(k+1)}$ and $k$.
Moreover, for $i<k$, it holds
\begin{equation*}
\begin{split}
|D_{k,i}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})|&\le \frac{\alpha}{\Gamma(1-\alpha)}C^{(k,i)}\Big((\Delta t)^{k-\alpha}\sum_{j=1}^{k-i}\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\mathrm{d}s \\
&+(\Delta t)^{k+1-\alpha}\sum_{j=1}^{n}\int_{0}^{1}(n-j+1-s)^{-\alpha-1}(1-s)\mathrm{d}s\Big) \\
&\le C^{(k,i)}\Big(\frac{\alpha}{\Gamma(1-\alpha)}(\Delta t)^{k+1}(k-i)(t_{n-k+i})^{-\alpha-1} \\
&+(\Delta t)^{k+1-\alpha}\big(\frac{1}{\Gamma(1-\alpha)}+\frac{1}{\Gamma(2-\alpha)}\big)\Big),
\end{split}
\end{equation*}
where $C^{(k,i)}$ is a constant depending on $u^{(k)}$, $u^{(k+1)}$ and $k, i$.
\end{proof} | 3,150 | 54,748 | en |
train | 0.45.5 | \begin{remark}
It is shown from formula \eqref{eq:truncki} that the order accuracy isn't uniform for all $n\ge k$. In the case of $t_{n}$ being near the origin, the accuracy order of the local truncation error reduced to the $(k-\alpha)$ order, in view that the $(k-1)$ degree polynomials as shown in \eqref{eq:P23} are chosen on the subinterval $\cup_{j=1}^{k-i} I_{j}$. However, replacing polynomials of degree $k$ on the corresponding subinterval can avoid this drawback, which is shown in \eqref{eq:trunckk}.
\end{remark}
\begin{remark}
There is need to point out that the local truncation error estimations \eqref{eq:truncki} and \eqref{eq:trunckk} holds only in the case of the solution $u(t)$ possessing proper smoothness on the closed interval $[0, T]$. In order to check the convergence rate of the global error when $f(t, u(t))$ is smooth with respect to $t$ and $u$, we apply the methods \eqref{Dki1} in the cases of $1\le i\le k\le 3$ on the test equation
\begin{equation}\label{eq:duf}
{^{C}}D^{\alpha}u(t)=f(t),\hspace{0.618cm} t\in (0, 1],
\end{equation}
such that the exact solution is $u(t)=E_{\alpha,1}(-t^{\alpha})\in C[0,1]\cap C^{\infty}(0,1]$. In Table \ref{ta:1} and \ref{ta:2}, the accuracy and the convergence order of the error $|u(t_{M})-u_{M}|$ are shown for different timestep and order $\alpha$. According to the numerical experiment, the high order convergence seems to reduce to the first order in the cases of $1\le i\le k\le 3$.
It is because that the solution of the problem \eqref{eq:nolinfode} only possess continuity on interval $I$ if function $f(t, u(t))$ is smooth on $I$. On the other hand, the integer order derivative of any smooth function on a compact domain $I$ still preserves to be smooth on $I$, in contrast, the $\alpha$ order fractional derivative of the smooth function isn't smooth any more, which implies there exist some continuous functions $f(t,u)$ such that the solution is smooth on $I$.
\end{remark}
\begin{table}
\centering
\caption{The error accuracy and convergence rate of $|u(t_{M})-u_{M}|$ in problem \eqref{eq:duf}.}
\label{ta:1}
\footnotesize
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}l@{\extracolsep{1cm}}l @{\extracolsep{1.4cm}}l@{\extracolsep{1cm}}l@{\extracolsep{1cm}} l@{\extracolsep{1cm}}l @{\extracolsep{1cm}}l@{\extracolsep{1cm}}l@{\extracolsep{\fill}} }
\toprule
$\alpha$& $M$ & \multicolumn{2}{l}{$(k, i)=(1,1)$} & \multicolumn{2}{l}{$(k,i)=(2, 1)$}&\multicolumn{2}{l}{$(k,i)=(2, 2)$} \\
\cline{3-8}
&&$|u(t_{M})-u_{M}|$&rate&$|u(t_{M})-u_{M}|$ &rate &$|u(t_{M})-u_{M}|$ &rate \\
\midrule
0.1 & 20 & 4.70994E-03 & - & 9.55476E-04 & - & 9.67511E-04 & - \\
& 40 & 2.37818E-03 & 0.99 & 4.93426E-04 & 0.95 & 4.96390E-04 & 0.96 \\
& 80 & 1.21118E-03 & 0.97 & 2.53608E-04 & 0.96 & 2.54352E-04 & 0.96 \\
& 160 & 6.19034E-04 & 0.97 & 1.30008E-04 & 0.96 & 1.30197E-04 & 0.97 \\
& 320 & 3.16763E-04 & 0.97 & 6.65299E-05 & 0.97 & 6.65780E-05 & 0.97 \\
\midrule
0.5 & 20 & 3.59879E-02 & - & 1.27599E-03 & - & 1.24693E-03 & - \\
& 40 & 1.87445E-02 & 0.94 & 5.84522E-04 & 1.13 & 5.76449E-04 & 1.11 \\
& 80 & 9.67807E-03 & 0.95 & 2.79573E-04 & 1.06 & 2.77440E-04 & 1.06 \\
& 160 & 4.95832E-03 & 0.96 & 1.36716E-04 & 1.03 & 1.36166E-04 & 1.03 \\
& 320 & 2.52435E-03 & 0.97 & 6.76067E-05 & 1.02 & 6.74666E-05 & 1.01 \\
\midrule
0.9 & 20 & 6.28955E-02 & - & 2.95957E-03 & - & 2.96453E-03 & - \\
& 40 & 3.23694E-02 & 0.96 & 1.33551E-03 & 1.15 & 1.33646E-03 & 1.15 \\
& 80 & 1.64619E-02 & 0.98 & 6.25412E-04 & 1.09 & 6.25601E-04 & 1.10 \\
& 160 & 8.31769E-03 & 0.98 & 3.00737E-04 & 1.06 & 3.00775E-04 & 1.06 \\
& 320 & 4.18769E-03 & 0.99 & 1.47049E-04 & 1.03 & 1.47057E-04 & 1.03 \\
\bottomrule
\end{tabular*}
\end{table}
\begin{table}[ht]
\centering
\footnotesize
\caption{The error accuracy and convergence rate of $|u(t_{M})-u_{M}|$ in problem \eqref{eq:duf}.}
\label{ta:2}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}l@{\extracolsep{1cm}}l @{\extracolsep{1.4cm}}l@{\extracolsep{1cm}}l@{\extracolsep{1cm}} l@{\extracolsep{1cm}}l @{\extracolsep{1cm}}l@{\extracolsep{1cm}}l@{\extracolsep{\fill}} }
\toprule
$\alpha$& $M$ & \multicolumn{2}{l}{$(k, i)=(3,1)$} & \multicolumn{2}{l}{$(k,i)=(3,2)$}&\multicolumn{2}{l}{$(k,i)=(3,3)$} \\
\cline{3-8}
&&$|u(t_{M})-u_{M}|$&rate&$|u(t_{M})-u_{M}|$ &rate &$|u(t_{M})-u_{M}|$ &rate \\
\midrule
0.1 & 20 & 9.24206E-04 & - & 9.17788E-04 & - & 9.23060E-04 & - \\
& 40 & 4.71821E-04 & 0.97 & 4.70329E-04 & 0.96 & 4.71636E-04 & 0.97 \\
& 80 & 2.41265E-04 & 0.97 & 2.40900E-04 & 0.97 & 2.41229E-04 & 0.97 \\
& 160 & 1.23379E-04 & 0.97 & 1.23288E-04 & 0.97 & 1.23371E-04 & 0.97 \\
& 320 & 6.30607E-05 & 0.97 & 6.30376E-05 & 0.97 & 6.30591E-05 & 0.97 \\
\midrule
0.5 & 20 & 2.87149E-03 & - & 2.85982E-03 & - & 2.86650E-03 & - \\
& 40 & 1.42015E-03 & 1.02 & 1.41730E-03 & 1.01 & 1.41912E-03 & 1.01 \\
& 80 & 7.06991E-04 & 1.01 & 7.06289E-04 & 1.00 & 7.06757E-04 & 1.01 \\
& 160 & 3.52818E-04 & 1.00 & 3.52644E-04 & 1.00 & 3.52763E-04 & 1.00 \\
& 320 & 1.76251E-04 & 1.00 & 1.76208E-04 & 1.00 & 1.76237E-04 & 1.00 \\
\midrule
0.9 & 20 & 1.30920E-03 & - & 1.30931E-03 & - & 1.30919E-03 & - \\
& 40 & 6.18111E-04 & 1.08 & 6.18099E-04 & 1.08 & 6.18098E-04 & 1.08 \\
& 80 & 3.00274E-04 & 1.04 & 3.00268E-04 & 1.04 & 3.00270E-04 & 1.04 \\
& 160 & 1.47969E-04 & 1.02 & 1.47967E-04 & 1.02 & 1.47968E-04 & 1.02 \\
& 320 & 7.34347E-05 & 1.01 & 7.34342E-05 & 1.01 & 7.34345E-05 & 1.01 \\
\bottomrule
\end{tabular*}
\end{table} | 3,167 | 54,748 | en |
train | 0.45.6 | \section{Stability analysis}\label{stabsection}
To consider the numerical stability of schemes \eqref{eq:nnonliode} with initial value $u(0)=u_{0}$,
the analysis on the linear difference equation
\begin{equation}
\label{DkiTest}
D_{k,i}^{\alpha}u_{n}=\lambda u_{n}, \hspace{0.618cm} n\ge k
\end{equation}
is given as follows. Using formulae \eqref{DkiTest}, we construct the equivalent relationship with respect to the generating power series
\begin{equation*}
\sum_{n=0}^{\infty}D_{k,i}^{\alpha}u_{n+k}\xi^{n}=\lambda\sum_{n=0}^{\infty}u_{n+k}\xi^{n}.
\end{equation*}
Replacing \eqref{Dki1}, one hence has
\begin{equation}
\label{DkiTest1}
\omega^{(k,i)}(\xi)u(\xi)=zu(\xi)+g^{(k,i)}(\xi),
\end{equation}
where $z:=\lambda(\Delta t)^{\alpha}$, the formal power series are denoted by
\begin{equation}\label{eq:fps}
\begin{split}
& u(\xi)=\sum\limits_{n=0}^{\infty}u_{n+k}\xi^{n}, \hspace{0.2cm} \omega^{(k,i)}(\xi)=\sum_{n=0}^{\infty}\omega_{n}^{(k,i)}\xi^{n}, \\
&g^{(k,i)}(\xi)=-\sum_{j=0}^{k-1}u_{j}\sum_{n=0}^{\infty}\big(w_{n+k,j}^{(k,i)}+\omega_{n+k-j}^{(k,i)}\big)\xi^{n}.
\end{split}
\end{equation}
Inspired by stability anlyses in \cite{LubichC:1983,LubichC:1986b}, we therefore present the following preliminary conclusions.
\begin{lemma}[\cite{LubichC:1986b}]
\label{le:3.4}
Assume that the coefficient sequence of $a(\xi)$ is in $l^{1}$. Let $|\xi_{0}|\le 1$. Then the coefficient sequence of
\begin{equation*}
b(\xi)=\frac{a(\xi)-a(\xi_{0})}{\xi-\xi_{0}}
\end{equation*}
converges to zero.
\end{lemma}
\begin{theorem}[\cite{Rudin:1987, ZygmundA:2002}]
\label{th:pw}
Suppose that
\begin{equation*}
f(z)=\sum_{n=0}^{\infty}c_{n}z^{n}, \hspace{0.4cm} \sum_{n=0}^{\infty}|c_{n}|<\infty,
\end{equation*}
and $f(z)\ne 0$ for every $|z|\le 1$. Then
\begin{equation*}
\frac{1}{f(z)}=\sum_{n=0}^{\infty}a_{n}z^{n}\hspace{0.2cm}\text{with}\hspace{0.1cm} \sum_{n=0}^{\infty}|a_{n}|<\infty.
\end{equation*}
\end{theorem}
\begin{theorem}[\cite{AkhiezerN:1965, ShohatJJ:1970}]
\label{th:cm}
For the moment problem
\begin{equation*}
s_{k}=\int_{0}^{1}u^{k}\mathrm{d}\sigma(u),\hspace{0.5cm} k=0, 1, \cdots
\end{equation*}
to be soluble within the class of non-decreasing functions iff the inequalities
\begin{equation*}
(-1)^{m}\nabla^{m}s_{k}\ge 0
\end{equation*}
hold for $k\ge m$.
\end{theorem}
\begin{lemma}
\label{le:3.1}
The coefficient sequences of series $g^{(k,i)}(\xi)$ converge to zero.
\end{lemma}
\begin{proof}
According to the expression of $\nabla^{k}I_{n,q}^{r}$ in Lemma \ref{le:Ijqr}, it yields that
\begin{equation*}
\lim_{n\to\infty}\nabla^{k}I_{n,q}^{r}=\frac{(-\alpha)_{k}}{\Gamma(1-\alpha)}\int_{[0,1]^{k+1}}\lim_{n\to\infty}\Big(\sum_{i=1}^{k}\xi_{i}+n-k+1-s\Big)^{-\alpha-k}\mathrm{d}^{k}\mathbf{\xi}\mathrm{d}\binom{s-q+r-1}{r}=0
\end{equation*}
or
\begin{equation*}
\lim_{n\to\infty}\nabla^{k}I_{n,q}^{r}=\frac{(-\alpha)_{k+1}}{\Gamma(1-\alpha)}\int_{[0, 1]^{k+1}}\lim_{n\to\infty}\Big(\sum_{i=1}^{k}\xi_{i}+n-k+1-s\Big)^{-\alpha-k-1}\binom{s-q+r-1}{r}\mathrm{d}^{k}\mathbf{\xi}\mathrm{d}s=0
\end{equation*}
for $k, q, r\in\mathbb{N}^{+}$ that are independent of $n$ and $\alpha>0$. Note that $g_{n}^{(k,i)}=-\sum\limits_{j=0}^{k-1}u_{j}(w_{n+k,j}^{(k,i)}+\omega_{n+k-j}^{(k,i)})$ is the finite linear combination of $\nabla^{k}I_{j,q}^{r}$ for finite $k$, thus it deduces $g_{n}^{(k,i)}\to 0$ as $n\to\infty$ if $\{u_{j}\}_{j=0}^{k-1}$ are bounded.
\end{proof}
\begin{lemma}
\label{le:3.2}
For $1\le i\le k\le 6$, the coefficient sequence of $\omega^{(k,i)}(\xi)$ belongs to $l^{1}$ space.
\end{lemma}
\begin{proof}
As indicated in Lemma \ref{le:Ijqr} and Lemma \ref{le:3.1}, the following relationship
\begin{equation}
\label{eq:3.3a}
\sum_{n=p}^{\infty}|\nabla^{k}I_{n,q}^{r}|=|\sum_{n=p}^{\infty} (\nabla^{k-1}I_{n,q}^{r}-\nabla^{k-1}I_{n-1,q}^{r})|=|\nabla^{k-1}I_{p-1,q}^{r}|
\end{equation}
holds for $p\ge k\ge 1$. Therefore, according to the definition of sequence $\{\omega_{n}^{(k,i)}\}_{n=0}^{\infty}$, there exists finite positive integer $M=M(k,i)$, such that
\begin{equation*}
\begin{split}
\sum_{n=0}^{\infty}|\omega_{n}^{(k,i)}|&\le \sum_{n=0}^{M}|\omega_{n}^{(k,i)}|+\sum_{m=1}^{k}\sum_{n=m}^{\infty}|\nabla^{m}I_{n,i}^{m}| \\
&\le \sum_{n=0}^{M}|\omega_{n}^{(k,i)}|+\sum_{m=1}^{k}|\nabla^{m-1} I_{m-1,i}^{m}|,
\end{split}
\end{equation*}
which implies the result.
\end{proof}
\begin{lemma}
\label{le:3.3}
For $1\le i\le k\le 6$ and $|\xi_{0}|\le 1$, the coefficient sequence of $(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}$ belongs to $l^{1}$ space.
\end{lemma}
\begin{proof}
According to the expression of $\omega^{(k,i)}(\xi)$, the following series can be rewritten to
\begin{equation*}
\begin{split}
(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}&=(1-\xi)\sum_{n=0}^{\infty}\omega_{n}^{(k,i)}\frac{\xi^{n}-\xi_{0}^{n}}{\xi-\xi_{0}} \\
&=(1-\xi)\sum_{n=1}^{\infty}\omega_{n}^{(k,i)}\sum_{m=0}^{n-1}\xi_{0}^{n-1-m}\xi^{m} \\
&=(1-\xi)\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\omega_{n+m+1}^{(k,i)}\xi_{0}^{n}\xi^{m} \\
&=\sum_{n=0}^{\infty}\omega_{n+1}^{(k,i)}\xi_{0}^{n}+\sum_{m=1}^{\infty}\big(\sum_{n=0}^{\infty}\nabla \omega_{n+m+1}^{(k,i)}\xi_{0}^{n}\big)\xi^{m}.
\end{split}
\end{equation*}
On one hand, from Lemma \ref{le:3.2}, we have
\begin{equation*}
|\sum_{n=0}^{\infty}\omega_{n+1}^{(k,i)}\xi_{0}^{n}|\le \sum_{n=0}^{\infty}|\omega_{n+1}^{(k,i)}||\xi_{0}|^{n}\le \sum_{n=0}^{\infty}|\omega_{n+1}^{(k,i)}|<+\infty.
\end{equation*}
On the other hand, by the definition of $\{\nabla^{k+1}I_{n,q}^{r}\}_{n=k+1}^{\infty}$ in Lemma \ref{le:Ijqr}, it can be verified that
\begin{equation*}
\begin{split}
\sum_{m=p}^{\infty}\sum_{n=0}^{\infty}|\nabla^{k+1} I_{m+n+1,q}^{r}|&=|\sum_{m=p}^{\infty}\sum_{n=0}^{\infty}\big(\nabla^{k} I_{m+n+1,q}^{r}-\nabla^{k} I_{m+n,q}^{r}\big)| \\
&=|\sum_{m=p}^{\infty}\big(\nabla^{k-1} I_{m,q}^{r}-\nabla^{k-1} I_{m-1,q}^{r}\big)| \\
&=|\nabla^{k-1}I_{p-1,q}^{r}|
\end{split}
\end{equation*}
for $p\ge k\ge 1$. Therefore there exists $M_{1}=M_{1}(k,i)\ge 1$ and $M_{2}=M_{2}(k,i)\ge 0$ such that
\begin{equation*}
\begin{split}
\sum_{m=1}^{\infty}\sum_{n=0}^{\infty}|\nabla \omega_{n+m+1}^{(k,i)}|&\le \sum_{m=1}^{M_{1}}\sum_{n=0}^{M_{2}}|\nabla \omega_{n+m+1}^{(k,i)}|+\sum_{p=1}^{k}\sum_{m=p}^{\infty}\sum_{n=0}^{\infty}|\nabla^{p+1}I_{m+n+1,i}^{p}| \\
&\le \sum_{m=1}^{M_{1}}\sum_{n=0}^{M_{2}}|\nabla \omega_{n+m+1}^{(k,i)}|+\sum_{p=1}^{k}|\nabla^{p-1}I_{p-1,i}^{p}|.
\end{split}
\end{equation*}
Combining with
\begin{equation*}
|\sum_{n=0}^{\infty}\omega_{n+1}^{(k,i)}\xi_{0}^{n}|+\sum_{m=1}^{\infty}|\sum_{n=0}^{\infty}\nabla \omega_{n+m+1}^{(k,i)}\xi_{0}^{n}|\le \sum_{n=0}^{\infty}|\omega_{n+1}^{(k,i)}|+\sum_{m=1}^{\infty}\sum_{n=0}^{\infty} |\nabla \omega_{n+m+1}^{(k,i)}|,
\end{equation*}
we arrive at the conclusion.
\end{proof}
\begin{corollary}
\label{co:1}
For any $|\xi_{0}|\le 1$ and $1\le i\le k\le 6$, it holds that the sequence
\begin{equation*}
(1-\xi)(1-\xi_{0})\frac{\varphi^{(k,i)}(\xi)-\varphi^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}
\end{equation*}
belongs to $l^{1}$ space, where series $\varphi^{(k,i)}(\xi)$ is defined to satisfy the relation $
\omega^{(k,i)}(\xi)=(1-\xi)\varphi^{(k,i)}(\xi)$.
\end{corollary}
\begin{proof}
Based on the definition of $\varphi^{(k,i)}(\xi)$, it yields that
\begin{equation*}
\begin{split}
(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}&=(1-\xi)\frac{(1-\xi)\varphi^{(k,i)}(\xi)-(1-\xi_{0})\varphi^{(k,i)}(\xi_{0})}{\xi-\xi_{0}} \\
&=(1-\xi)(1-\xi_{0})\frac{\varphi^{(k,i)}(\xi)-\varphi^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}-\omega^{(k,i)}(\xi),
\end{split}
\end{equation*}
because of the absolute convergence of sequences $(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}$ and $\omega^{(k,i)}(\xi)$ shown in Lemma \ref{le:3.2} and Lemma \ref{le:3.3}, we shall arrive at the result.
\end{proof} | 3,495 | 54,748 | en |
train | 0.45.7 | \begin{corollary}
\label{co:1}
For any $|\xi_{0}|\le 1$ and $1\le i\le k\le 6$, it holds that the sequence
\begin{equation*}
(1-\xi)(1-\xi_{0})\frac{\varphi^{(k,i)}(\xi)-\varphi^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}
\end{equation*}
belongs to $l^{1}$ space, where series $\varphi^{(k,i)}(\xi)$ is defined to satisfy the relation $
\omega^{(k,i)}(\xi)=(1-\xi)\varphi^{(k,i)}(\xi)$.
\end{corollary}
\begin{proof}
Based on the definition of $\varphi^{(k,i)}(\xi)$, it yields that
\begin{equation*}
\begin{split}
(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}&=(1-\xi)\frac{(1-\xi)\varphi^{(k,i)}(\xi)-(1-\xi_{0})\varphi^{(k,i)}(\xi_{0})}{\xi-\xi_{0}} \\
&=(1-\xi)(1-\xi_{0})\frac{\varphi^{(k,i)}(\xi)-\varphi^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}-\omega^{(k,i)}(\xi),
\end{split}
\end{equation*}
because of the absolute convergence of sequences $(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}$ and $\omega^{(k,i)}(\xi)$ shown in Lemma \ref{le:3.2} and Lemma \ref{le:3.3}, we shall arrive at the result.
\end{proof}
\begin{theorem}
\label{the:stabilityr}
The stability region of method $D_{k,i}^{\alpha}u_{n}=\lambda u_{n}$ is $\mathrm{S}^{(k,i)}=\mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$ in the cases of $1\le i\le k\le 6$.
\end{theorem}
\begin{remark}
The definition of stability region $\mathrm{S}^{(k,i)}$ of method $D_{k,i}^{\alpha}u_{n}=\lambda u_{n}$ is the set of $z=\lambda (\Delta t)^{\alpha}\in\mathbb{C}$ with $\Delta t>0$ for which there is $u_{n}\to 0$ as $n\to\infty$ whenever the starting values $u_{0}, \cdots, u_{k-1}$ are bounded.
\end{remark}
\begin{proof}
The provement of $S^{(k,i)}=\mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$ is equivalent with proving both $S^{(k,i)}\supseteq\mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$ and $S^{(k,i)}\subseteq\mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$, i.e., to prove that for any $z\in \mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$, there is $z\in S^{(k,i)}$ and for any $z\not \in \mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$, there is $z\not\in S^{(k,i)}$.
On one hand, if $z\in \mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$ and $|z|\le 1$, there is $z-\omega^{(k,i)}(\xi)\neq 0$ for $|\xi|\le 1$, thus according to Lemma \ref{le:3.1}, Lemma \ref{le:3.2} and Theorem \ref{th:pw}, it yields that the coefficient sequence of reciprocal of $z-\omega^{(k,i)}(\xi)$ is in $l^{1}$ and coefficient sequence of series $g^{(k,i)}(\xi)$ tends to zero.
If $|z|>1$, formula \eqref{DkiTest1} can be rewritten to
\begin{equation*}
u(\xi)=\frac{\frac{g^{(k,i)}(\xi)}{z}}{\frac{\omega^{(k,i)}(\xi)}{z}-1},
\end{equation*}
in which case the coefficient sequence of reciprocal of $\frac{\omega^{(k,i)}(\xi)}{z}-1$ is in $l^{1}$, and the coefficient sequence of series $\frac{g^{(k,i)}(\xi)}{z}$ converges to zero. In addition, assume that $\lim\limits_{n\to\infty}\sum\limits_{j=0}^{n}|l_{i}|=L<+\infty$ and $\lim\limits_{j\to\infty}c_{j}=0$, it holds that $\lim\limits_{n\to\infty}\sum\limits_{j=0}^{n}l_{n-j}c_{j}=0$, thus, implies that $u_{n}\to 0$ as $n\to\infty$.
On the other hand, assume that for any $z=\omega^{(k,i)}(\xi_{0})$ with $|\xi_{0}|\le 1$, according to \eqref{DkiTest1} the solution satisfies that
\begin{equation}
\label{eq:3.3b}
\left(\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})\right)u(\xi)=g^{(k,i)}(\xi).
\end{equation}
Note that method \eqref{Dki1} is exact for constant function, which leads to
\begin{equation*}
\sum_{j=0}^{k-1}w_{n,j}^{(k,i)}+\sum_{j=0}^{n}\omega_{n-j}^{(k,i)}=0,\hspace{0.5cm} n\ge k,
\end{equation*}
and a corresponding formal power series satisfies that
\begin{equation*}
\begin{split}
&\sum_{n=k}^{\infty}\left(\sum_{j=0}^{k-1}w_{n,j}^{(k,i)}+\sum_{j=0}^{n}\omega_{n-j}^{(k,i)}\right)\xi^{n-k} \\
=&\sum_{n=0}^{\infty}\left(\sum_{j=0}^{k-1}w_{n+k,j}^{(k,i)}+\sum_{j=0}^{n+k}\omega_{n+k-j}^{(k,i)}\right)\xi^{n} \\
=&\sum_{n=0}^{\infty}\sum_{j=0}^{k-1}\left( w_{n+k,j}^{(k,i)}+\omega_{n+k-j}^{(k,i)} \right)\xi^{n}+\frac{\omega^{(k,i)}(\xi)}{1-\xi}=0. \\
\end{split}
\end{equation*}
Assume that $u_{0}=\cdots=u_{k-1}\neq0$, then according to the expression of $g^{(k,i)}(\xi)$, it holds that
$
g^{(k,i)}(\xi)=u_{0}\frac{\omega^{(k,i)}(\xi)}{1-\xi}$. In the case of $\omega^{(k,i)}(\xi_{0})=0$, it yields that $u(\xi)=\frac{u_{0}}{1-\xi}$, which means that $u_{n}=u_{0}$ for any $n\in \mathbb{N}$. And for the rest case, there is
\begin{equation*}
u(\xi)(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}=u_{0}\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}+u_{0}\frac{\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}.
\end{equation*}
If assume that $u_{n}\to0$ as $n\to\infty$, since according to Lemma \ref{le:3.3}, the coefficient sequence of $(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}$ is in $l_{1}$, which derives that the coefficient sequence of $u(\xi)(1-\xi)\frac{\omega^{(1,1)}(\xi)-\omega^{(1,1)}(\xi_{0})}{\xi-\xi_{0}}$ tends to zero, in addition, according to Lemma \ref{le:3.4}, it yields that the coefficient sequence of $\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}$ converges to zero, however, the divergence of the coefficient sequence of $\frac{1}{\xi-\xi_{0}}$ for $|\xi_{0}|\le 1$ leads to the contradiction. Thus, it holds that there exist some nonzero bounded initial values $\{u_{i}\}_{i=0}^{k-1}$ such that $u_{n} \not\to 0$ as $n\to\infty$, which indicates that $z\not\in \mathrm{S}^{(k,i)}$.
\end{proof} | 2,243 | 54,748 | en |
train | 0.45.8 | According to the definition of $A(\theta)$-stability \cite{HairerWN:1991} in usual case, we define the $A(\theta)$-stability in the following sense of $0<\alpha<1$.
\begin{definition}
A method is said to be $A(\theta)$-stable for $\theta\in[0, \pi-\frac{\alpha\pi}{2})$, if the sector
\begin{equation*}
S_{\theta}=\{z: |\mathrm{arg}(-z)|\le \theta, \hspace{0.2cm}z\neq 0\}
\end{equation*}
is contained in the stability region.
\end{definition}
\begin{theorem}\label{th:Api}
The methods \eqref{DkiTest} are $A(\frac{\pi}{2})$-stable in the cases of $1\le i\le k\le 2$.
\end{theorem}
\begin{proof}
In view of the definition of $A(\theta)$-stability, in particular, when $\theta=\frac{\pi}{2}$, it suffices to prove that $\mathrm{S}_{\frac{\pi}{2}}\subseteq S^{(k,i)}$ for $1\le i\le k\le 2$, i.e., to prove $\omega^{(k,i)}(\xi)=0$ for some $|\xi|\le 1$ and $\mathrm{Re}(\omega^{(k,i)}(\xi))>0$ otherwise.
First of all, it can be readily checked that $\omega^{(k,i)}(1)=0$, which implies $0\not \in \mathrm{S}_{\frac{\pi}{2}}$. In the case of $(k,i)=(1,1)$, resulting from the expression of $\omega^{(1,1)}(\xi)$, there is
\begin{equation}
\label{eq:omega11}
\omega^{(1,1)}(\xi)=I_{0}+\sum_{j=1}^{\infty}\nabla I_{j}\xi^{j}=(1-\xi)I(\xi),
\end{equation}
where $I(\xi)=\sum\limits_{n=0}^{\infty}I_{n}\xi^{n}$.
Since, according to Lemma \ref{le:Ijqr} and Theorem \ref{th:cm}, we have
\begin{equation}
\label{eq:ln}
I_{n}=\int_{0}^{1}r^{n}\mathrm{d}\sigma(r), \hspace{0.4cm} n\in \mathbb{N},
\end{equation}
where $\sigma(r)$ is a non-decreasing function,
then suppose that $|\xi|<1$, substituting \eqref{eq:ln} into \eqref{eq:omega11} yields that
\begin{equation*}
\mathrm{Re}\Big(\omega^{(1,1)}(\xi)\Big)=\mathrm{Re}\Big((1-\xi)\sum_{n=0}^{\infty}\int_{0}^{1}r^{n}\mathrm{d}\sigma(r)\xi^{n}\Big)=\int_{0}^{1}\mathrm{Re}\Big(\frac{1-\xi}{1-r\xi}\Big)\mathrm{d}\sigma(r).
\end{equation*}
Let $\xi=|\xi|(\cos\theta+i\sin\theta)$, there is
\begin{equation*}
\frac{1-\xi}{1-r\xi}
=\frac{\left(1-(r+1)|\xi|\cos\theta+r|\xi|^{2}\right)+i\left((r-1)|\xi|\sin\theta\right)}{(1-r|\xi|\cos\theta)^{2}+(r|\xi|\sin\theta)^{2}},
\end{equation*}
and for $0\le r\le1$ and $|\xi|<1$, it holds that
\begin{equation*}
\begin{split}
&1-(r+1)|\xi|\cos\theta+r|\xi|^{2}\ge \min\left((1-|\xi|\cos\theta)^{2},1-|\xi|\cos\theta\right), \\
&1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}\le (1+r|\xi|)^{2}\le 4,
\end{split}
\end{equation*}
which arrives at
\begin{equation*}
\int_{0}^{1}\mathrm{Re}\left(\frac{1-\xi}{1-r\xi}\right)\mathrm{d}\sigma(r)
\ge\frac{\min\left((1-|\xi|\cos\theta)^{2},1-|\xi|\cos\theta\right)}{4}I_{0}.
\end{equation*}
In other case of $(k,i)=(2,1)$, from the definition of $\omega^{(2,1)}(\xi)$, it induces that
\begin{equation}\label{eq:fomega21}
\begin{split}
\omega^{(2,1)}(\xi)=&\sum_{n=0}^{\infty}\left(\nabla I_{n}+\nabla^{2}I_{n,1}^{2}\right)\xi^{n} \\
=&(1-\xi)I(\xi)+(1-\xi)^{2}I_{1}^{2}(\xi) \\
=&(1-\xi)\left(I(\xi)-2I_{1}^{2}(\xi)+(3-\xi)I_{1}^{2}(\xi)\right),
\end{split}
\end{equation}
where
\begin{equation*}
I(\xi)=\sum_{n=0}^{\infty}I_{n}\xi^{n},\hspace{0.7cm} I_{1}^{2}(\xi)=\sum_{n=0}^{\infty}I_{n,1}^{2}\xi^{n}.
\end{equation*}
According to Lemma \ref{le:Ijqr}, Corollary \ref{co:s} and Theorem \ref{th:cm},
there exist non-decreasing functions $\upsilon$ and $\gamma$, respectively, such that
\begin{equation}
\label{eq:l12}
I_{n}-2I_{n,1}^{2}=\int_{0}^{1}r^{n}\mathrm{d}\upsilon(r), \hspace{0.3cm}n=0,1,\cdots,
\end{equation}
and
\begin{equation}
\label{eq:ln12}
I_{n,1}^{2}=\int_{0}^{1}r^{n}\mathrm{d}\gamma(r),\hspace{0.5cm}n=0,1,\cdots.
\end{equation}
Then for $|\xi|<1$, in place of $\omega^{(2,1)}(\xi)$, we can get
\begin{equation*}
\mathrm{Re}\Big(\omega^{(2,1)}(\xi)\Big)=\int_{0}^{1}\mathrm{Re}\Big(\frac{1-\xi}{1-r\xi}\Big)\mathrm{d}\upsilon(r)+\int_{0}^{1}\mathrm{Re}\Big(\frac{(1-\xi)(3-\xi)}{1-r\xi}\Big)\mathrm{d}\gamma(r).
\end{equation*}
Moreover, it indicates
\begin{equation*}
\begin{split}
\frac{(1-\xi)(3-\xi)}{1-r\xi}
=&\frac{(3-4|\xi|\cos\theta+|\xi|^{2}\cos2\theta)(1-r|\xi|\cos\theta)+(4-2|\xi|\cos\theta)r|\xi|^{2}\sin^{2}\theta}{(1-r|\xi|\cos\theta)^{2}+(r|\xi|\sin\theta)^{2}} \\
&\hspace{0.5cm}+i\frac{(3r-|\xi|^{2}r-4+2|\xi|\cos\theta)|\xi|\sin\theta}{(1-r|\xi|\cos\theta)^{2}+(r|\xi|\sin\theta)^{2}},
\end{split}
\end{equation*}
since
\begin{equation*}
\begin{split}
3-4|\xi|\cos\theta+|\xi|^{2}\cos2\theta=3-4|\xi|\cos\theta+2|\xi|^{2}\cos^{2}\theta-|\xi|^{2}\ge 2(1-|\xi|\cos\theta)^{2},
\end{split}
\end{equation*}
there is
\begin{equation*}
\int_{0}^{1}\mathrm{Re}\Big(\frac{(1-\xi)(3-\xi)}{1-r\xi}\Big)\mathrm{d}\gamma(r)
\ge\frac{\min\Big((1-|\xi|\cos\theta)^{3},(1-|\xi|\cos\theta)^{2}\Big)}{2}I_{0,1}^{2}.
\end{equation*}
For the rest case of $(k,i)=(2,2)$, we begin with the equivalent form of $\omega^{(2,2)}(\xi)$, which satisfies that
\begin{equation}\label{eq:fomega22}
\begin{split}
\omega^{(2,2)}(\xi)=&I_{0}(1-\xi)+I_{0,1}^{2}(1-\xi)^{2}+(1-\xi)\sum_{n=0}^{\infty}I_{n+1}\xi^{n}+(1-\xi)^{2}\sum_{n=0}^{\infty}I_{n+1,2}^{2}\xi^{n} \\
=&I_{0,1}^{2}(1-\xi)(3-\xi)+(1-\xi^{2})\sum_{n=0}^{\infty}I_{n+1,1}^{2}\xi^{n}+(1-\xi)\left(I(\xi)-2I_{1}^{2}(\xi)\right),
\end{split}
\end{equation}
since for any $n\ge 0$, because of the relation $I_{n}+I_{n,2}^{2}=I_{n,1}^{2}$, there is
\begin{equation*}
\begin{split}
&(1-\xi)\sum_{n=0}^{\infty}I_{n+1}\xi^{n}+(1-\xi)^{2}\sum_{n=0}^{\infty}I_{n+1,2}^{2}\xi^{n} \\
=&(1-\xi)\Big(\sum_{n=0}^{\infty}I_{n+1,1}^{2}\xi^{n}-\xi\sum_{n=0}^{\infty}I_{n+1,2}^{2}\xi^{n}\Big) \\
=&(1-\xi^{2})\sum_{n=0}^{\infty}I_{n+1,1}^{2}\xi^{n}+(1-\xi)\sum_{n=0}^{\infty}\left(I_{n+1}-2I_{n+1,1}^{2}\right)\xi^{n+1}\\
=&(1-\xi^{2})\sum_{n=0}^{\infty}I_{n+1,1}^{2}\xi^{n}+(1-\xi)\left(I(\xi)-2I_{1}^{2}(\xi)-(I_{0}-2I_{0,1}^{2})\right).
\end{split}
\end{equation*}
Consequently, suppose that $|\xi|<1$, substituting conclusions \eqref{eq:l12} and \eqref{eq:ln12} into $\omega^{(2,2)}(\xi)$, we have
\begin{equation*}
\begin{split}
\mathrm{Re}\Big(\omega^{(2,2)}(\xi)\Big)=&\int_{0}^{1}\mathrm{Re}\Big((1-\xi)(3-\xi)\Big)\mathrm{d}\gamma(r) \\
&+\int_{0}^{1}r\mathrm{Re}\Big(\frac{1-\xi^{2}}{1-r\xi}\Big)\mathrm{d}\gamma(r)+\int_{0}^{1}\mathrm{Re}\Big(\frac{1-\xi}{1-r\xi}\Big)\mathrm{d}\upsilon(r),
\end{split}
\end{equation*}
furthermore, there is
\begin{equation*}
\begin{split}
\frac{1-\xi^{2}}{1-r\xi}
=&\frac{(1-|\xi|^{2}\cos2\theta)(1-r|\xi|\cos\theta)+r|\xi|^{3}\sin\theta\sin2\theta}{(1-r|\xi|\cos\theta)^{2}+(r|\xi|\sin\theta)^{2}} \\
&+i\frac{(1-|\xi|^{2}\cos2\theta)r|\xi|\sin\theta-(1-r|\xi|\cos\theta)\rho^{2}\sin2\theta}{(1-r|\xi|\cos\theta)^{2}+(r|\xi|\sin\theta)^{2}}.
\end{split}
\end{equation*}
Since for $0\le r\le 1$, it holds that
\begin{equation*}
\begin{split}
&(1-|\xi|^{2}\cos2\theta)(1-r|\xi|\cos\theta)+r|\xi|^{3}\sin\theta\sin2\theta \\
=&1-|\xi|^{2}\cos2\theta-r|\xi|\cos\theta+r|\xi|^{3}\cos\theta \\
\ge&(1-|\xi|^{2})(1-|\xi||\cos\theta|),
\end{split}
\end{equation*}
then, we may see that
\begin{equation*}
\begin{split}
\int_{0}^{1}r\mathrm{Re}\Big(\frac{1-\xi^{2}}{1-r\xi}\Big)\mathrm{d}\gamma(r)
\ge& \frac{(1-|\xi|^{2})(1-|\xi||\cos\theta|)}{4}\int_{0}^{1}r\mathrm{d}\gamma(r) \\
=&\frac{(1-|\xi|^{2})(1-|\xi||\cos\theta|)}{4}I_{1,1}^{2}.
\end{split}
\end{equation*}
As a result, for $1\le i\le k\le 2$, it demonstrates that
\begin{equation*}
\mathrm{Re}\Big(\omega^{(k,i)}(\xi)\Big)\ge\frac{\min\left((1-|\xi|\cos\theta)^{2}, 1-|\xi|\cos\theta\right)}{4}I_{0}>0,\hspace{0.6cm}|\xi|<1.
\end{equation*}
In addition, according to Lemma \ref{le:3.3}, there exists constant $M^{(k,i)}>0$ such that
\begin{equation*}
|\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})| \le \frac{M^{(k,i)}}{|1-\xi |} |\xi-\xi_{0}|,\hspace{0.5cm} \xi\ne 1,
\end{equation*}
which yields the pointwise continuity of $\omega^{(k,i)}(\xi)$ for $|\xi|\le 1$ with the exception of $\xi=1$. Therefore for any fixed $\xi$ lying on the unit circle, the angle of which satisfying $\mathrm{arg}(\xi)=\theta_{\xi}\neq 0$, correspondingly, there exists a sequence $\xi_{n}=(1-\frac{1}{n})\xi$ with $|\xi_{n}|<1$ for any $n=1, 2, \cdots$, such that
\begin{equation*}
\mathrm{Re}\Big(\omega^{(k,i)}(\xi)\Big)=\lim\limits_{n\to\infty}\mathrm{Re}\Big(\omega^{(k,i)}(\xi_{n})\Big)\ge \frac{I_{0}}{4}\min\left( (1-\cos\theta_{\xi})^{2}, 1-\cos\theta_{\xi}\right)>0.
\end{equation*}
\end{proof} | 3,724 | 54,748 | en |
train | 0.45.9 | Theorem \ref{th:Api} shows that all the rest zeros of series $\omega^{(k,i)}(\xi)~(1\le i\le k\le2)$ are outside the unit disc besides $\xi=1$, followed by which we next consider the location of zeros of series $\omega^{(k,i)}(\xi)$ in the examples of $1\le i\le k\le 3$ and confirm that $\xi=1$ is a simple zero. The following result can be considered as a generalisation of the strong root condition.
\begin{theorem}
\label{coro:1}
For $1\le i\le k\le 3$, the series $\omega^{(k,i)}(\xi)$ satisfies the following conditions:
\begin{description}
\item[i).] $\omega^{(k,i)}(\xi)\ne 0$ within the unit circle $|\xi|\le 1$ and $\xi\ne 1$;
\item[ii).] $\xi=1$ is the simple zero.
\end{description}
\end{theorem}
\begin{proof}
It can be easily checked that $\omega^{(k,i)}(1)=0$, yielding that $\xi=1$ is a zero, if we rewrite the series $\omega^{(k,i)}(\xi)$ in the form of
\begin{equation}\label{eq:fomega}
\omega^{(k,i)}(\xi)=(1-\xi)\varphi^{(k,i)}(\xi),
\end{equation}
it remains to prove that $\varphi^{(k,i)}(\xi)\ne 0$ for $|\xi|\le 1$, which is suffice to prove that $\mathrm{Re}(\varphi^{(k,i)}(\xi))>0$ for all $|\xi|\le 1$. First of all, assuming $\xi=|\xi|e^{i\theta}$
with $|\xi|<1$. Then in the case of $(k,i)=(1,1)$, the rewritten form \eqref{eq:ln} deduces that
\begin{equation*}
\varphi^{(1,1)}(\xi)=I(\xi)=\int_{0}^{1}\frac{1}{1-r\xi}\mathrm{d\sigma(r)},
\end{equation*}
and furthermore,
\begin{equation*}
\mathrm{Re}\left(\varphi^{(1,1)}(\xi)\right)=\int_{0}^{1}\mathrm{Re}\left(\frac{1}{1-r\xi}\right)\mathrm{d\sigma(r)}:=\int_{0}^{1}f(r,|\xi|,\theta)\mathrm{d\sigma(r)},
\end{equation*}
where $f(r,|\xi|,\theta)=\frac{1-r|\xi|\cos\theta}{1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}}$. Since a calculation deduces
\begin{equation*}
\frac{\partial f}{\partial \theta}(r,|\xi|,\theta)=\frac{-r|\xi|\sin\theta (1-r^{2}|\xi|^{2})}{(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2})^{2}}
\end{equation*}
possesses the same sign as $(-\sin\theta)$ for $0\le r\le 1$ and $0\le |\xi|<1$, it hence follows that $0<f(r,|\xi|,\pi)\le f(r,|\xi|,\theta)\le f(r,|\xi|,0)$ and thus
\begin{equation*}
\mathrm{Re}\left(\varphi^{(1,1)}(\xi)\right)\ge \int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d\sigma(r)}\ge \frac{I_{0}}{2}.
\end{equation*}
In the case of $(k,i)=(2,1)$, according to formulae \eqref{eq:fomega21}, \eqref{eq:l12} and \eqref{eq:ln12}, one obtains
\begin{equation*}
\begin{split}
\varphi^{(2,1)}(\xi)=&I(\xi)-2I_{1}^{2}(\xi)+(3-\xi)I_{1}^{2}(\xi) \\
=&\int_{0}^{1}\frac{1}{1-r\xi}\mathrm{d}\upsilon(r)+\int_{0}^{1}\frac{3-\xi}{1-r\xi}\mathrm{d}\gamma(r)
\end{split}
\end{equation*}
and thus
\begin{equation*}
\begin{split}
\mathrm{Re}\left(\varphi^{(2,1)}(\xi)\right)=&\int_{0}^{1}\mathrm{Re}\left(\frac{1}{1-r\xi}\right)\mathrm{d}\upsilon(r)+\int_{0}^{1}\mathrm{Re}\left(\frac{3-\xi}{1-r\xi}\right)\mathrm{d}\gamma(r) \\
\ge&\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\upsilon(r)+2\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\gamma(r)+\int_{0}^{1}\mathrm{Re}\left(\frac{1-\xi}{1-r\xi}\right)\mathrm{d}\gamma(r) >\frac{I_{0}}{2}.
\end{split}
\end{equation*}
In the case of $(k,i)=(2,2)$, it can be obtained from \eqref{eq:fomega22} that
\begin{equation*}
\varphi^{(2,2)}(\xi)=I_{0,1}^{2}(3-\xi)+(1+\xi)\sum_{n=0}^{\infty}I_{n+1,1}^{2}\xi^{n}+I(\xi)-2I_{1}^{2}(\xi),
\end{equation*}
and consequently, the real part of the series can be expressed by
\begin{equation*}
\mathrm{Re}\left(\varphi^{(2,2)}(\xi)\right)=I_{0,1}^{2}\mathrm{Re}(3-\xi)+\int_{0}^{1}r\mathrm{Re}\left(\frac{1+\xi}{1-r\xi}\right)\mathrm{d}\gamma(r)+\int_{0}^{1}\mathrm{Re}\left(\frac{1}{1-r\xi}\right)\mathrm{d}\upsilon(r)\ge\frac{I_{0}}{2}+I_{0,1}^{2}.
\end{equation*}
Since
\begin{equation*}
\mathrm{Re}\left(\frac{1+\xi}{1-r\xi}\right)=\frac{1-r|\xi|\cos\theta+|\xi|\cos\theta-r|\xi|^{2}}{1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}}:=f(r,|\xi|,\theta),
\end{equation*}
the relation $\frac{\partial f}{\partial \theta}(r,|\xi|,\theta)=\frac{-|\xi|\sin\theta(r+1)(1-r^{2}|\xi|^{2})}{(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2})^{2}}$ induces that $f(r,|\xi|,\theta)\ge f(r,|\xi|,\pi)>0$ for $0\le r\le 1$ and $0\le |\xi|<1$.
In the case of $(k,i)=(3,1)$, based on \eqref{eq:ki31}, the rewritten form of series $\omega^{(3,1)}(\xi)$ is
\begin{equation*}
\omega^{(3,1)}(\xi)=(1-\xi)I(\xi)+(1-\xi)^{2}I_{1}^{2}(\xi)+(1-\xi)^{3}I_{1}^{3}(\xi),
\end{equation*}
and consequently $\varphi^{(3,1)}(\xi)$ is given by
\begin{equation*}
\begin{split}
\varphi^{(3,1)}(\xi)&=I(\xi)+(1-\xi)I_{1}^{2}(\xi)+(1-\xi)^{2}I_{1}^{3}(\xi) \\
&=I(\xi)-3I_{1}^{3}(\xi)+(1-\xi)I_{1}^{2}(\xi)+(4-2\xi+\xi^{2})I_{1}^{3}(\xi). \\
\end{split}
\end{equation*}
According to Lemma \ref{le:1.5}, it yields that
\begin{equation*}
\begin{split}
I_{n}-3I_{n,1}^{3}&=\frac{3}{2}\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}(1-s^{2})\mathrm{d}s, \hspace{0.5cm} n\ge 0\\
\end{split}
\end{equation*}
is a completely monotonic sequence, from Theorem \ref{th:cm} there exists a non-decreasing function $\eta$ such that
\begin{equation*}
I_{n}-3I_{n,1}^{3}=\int_{0}^{1}r^{n}\mathrm{d}\eta(r), \hspace{0.5cm} n=0,1,\cdots.
\end{equation*}
In addition, we have from Lemma \ref{le:Ijqr} that sequence $\{I_{n,1}^{3}\}_{n=0}^{\infty}$ is a complete monotonic sequence, therefore it can be represented by
\begin{equation}\label{eq:ln13}
I_{n,1}^{3}=\int_{0}^{1}r^{n}\mathrm{d}\beta(r), \hspace{0.5cm} n=0,1,\cdots,
\end{equation}
where the function $\beta(r)$ is non-decreasing on $[0,1]$. We thus represent the series into the integral form and take the real part,
\begin{equation*}
\begin{split}
\mathrm{Re}\left(\varphi^{(3,1)}(\xi)\right)=&
\int_{0}^{1}\mathrm{Re}\left(\frac{1}{1-r\xi}\right)\mathrm{d}\eta(r)+\int_{0}^{1}\mathrm{Re}\left(\frac{1-\xi}{1-r\xi}\right)\mathrm{d}\gamma(r)+\int_{0}^{1}\mathrm{Re}\left(\frac{4-2\xi+\xi^{2}}{1-r\xi}\right)\mathrm{d}\beta(r) \\
\ge&\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\eta(r)+\frac{5}{2}\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\beta(r)
\ge\frac{1}{2}I_{0}-\frac{1}{4}I_{0,1}^{3},
\end{split}
\end{equation*}
since there holds following estimate
\begin{equation*}
\mathrm{Re}\left(\frac{\frac{3}{2}-2\xi+\xi^{2}}{1-r\xi}\right)=\frac{(1-r|\xi|\cos\theta)(\frac{3}{2}-2|\xi|\cos \theta+|\xi|^{2}\cos2\theta)+2r|\xi|^{2}\sin^{2}\theta(1-|\xi|\cos\theta)}{1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}},
\end{equation*}
and there is
\begin{equation*}
\frac{3}{2}-2|\xi|\cos \theta+|\xi|^{2}\cos2\theta=\frac{1}{2}(1-2|\xi|\cos\theta)^{2}+(1-|\xi|^{2})\ge 0
\end{equation*}
for $|\xi|\le 1$ and $\theta\in\mathbb{R}$.
In the case of $(k,i)=(3,2)$, according to the representation of $\{\omega_{n}^{(2,2)}\}_{n=0}^{\infty}$ in \eqref{eq:ki32}, we derive the expression of $\varphi^{(3,2)}(\xi)$ by
\begin{equation}\label{eq:phi32}
\begin{split}
\varphi^{(3,2)}(\xi)=&I_{0}+\sum_{j=0}^{\infty}I_{j+1}\xi^{j}+I_{0,1}^{2}(1-\xi)+(1-\xi)\sum_{j=0}^{\infty}I_{j+1,2}^{2}\xi^{j}+(1-\xi)^{2}I_{0,1}^{3}+(1-\xi)^{2}\sum_{j=0}^{\infty}I_{j+1,2}^{3}\xi^{j},
\end{split}
\end{equation}
Substituting the relations $
I_{n,2}^{2}=I_{n,1}^{2}-I_{n}$ and $
I_{n,2}^{3}=I_{n,1}^{3}-I_{n,1}^{2}$ into \eqref{eq:phi32} deduces that
\begin{equation*}
\begin{split}
\varphi^{(3,2)}(\xi)
=&I(\xi)+(1-\xi)I_{1}^{2}(\xi)+(1-\xi^{2})\sum_{j=0}^{\infty}
I_{j+1,1}^{3}\xi^{j} \\
&-2(1-\xi)I_{1}^{3}(\xi)+(3-\xi)(1-\xi)I_{0,1}^{3} \\
=&(\frac{5}{6}+\frac{1}{6}\xi)I(\xi)+(1-\xi)(I_{1}^{2}(\xi)-2I_{1}^{3}(\xi)+\frac{1}{6}I(\xi))
\\
&+(1-\xi^{2})\sum_{j=0}^{\infty}I_{j+1,1}^{3}\xi^{j}+(3-\xi)(1-\xi)I_{0,1}^{3}.
\end{split}
\end{equation*}
Based on Lemma \ref{le:1.5}, one hence obtains that
\begin{equation*}
I_{n,1}^{2}-2I_{n,1}^{3}+\frac{1}{6}I_{n}=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}s(1-s)\mathrm{d}s, \hspace{0.5cm}n\ge 0
\end{equation*}
is a completely monotonic sequence. Therefore the sequence can be expressed by
\begin{equation}\label{eq:lnmix32}
I_{n,1}^{2}-2I_{n,1}^{3}+\frac{1}{6}I_{n}=\int_{0}^{1}r^{n}\mathrm{d}\mu(r),\hspace{0.5cm} n=0,1,\cdots
\end{equation}
together with the function $\mu(r)$ being non-decreasing on interval $[0, 1]$. Therefore, from formulae \eqref{eq:ln}, \eqref{eq:lnmix32} and \eqref{eq:ln13}, it follows
\begin{equation*}
\begin{split}
\mathrm{Re}(\varphi^{(3,2)}(\xi))=&\int_{0}^{1}\mathrm{Re}(\frac{\frac{5}{6}+\frac{1}{6}\xi}{1-r\xi})\mathrm{d}\sigma(r)
+\int_{0}^{1}\mathrm{Re}(\frac{1-\xi}{1-r\xi})\mathrm{d}\mu(r) \\
+&\int_{0}^{1}r\mathrm{Re}(\frac{1-\xi^{2}}{1-r\xi})\mathrm{d}\beta(r)+\mathrm{Re}((3-\xi)(1-\xi))I_{0,1}^{3} \\
\ge& \frac{2}{3}\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\sigma(r)\ge \frac{I_{0}}{3}.
\end{split}
\end{equation*} | 3,818 | 54,748 | en |
train | 0.45.10 | In the case of $(k,i)=(3,2)$, according to the representation of $\{\omega_{n}^{(2,2)}\}_{n=0}^{\infty}$ in \eqref{eq:ki32}, we derive the expression of $\varphi^{(3,2)}(\xi)$ by
\begin{equation}\label{eq:phi32}
\begin{split}
\varphi^{(3,2)}(\xi)=&I_{0}+\sum_{j=0}^{\infty}I_{j+1}\xi^{j}+I_{0,1}^{2}(1-\xi)+(1-\xi)\sum_{j=0}^{\infty}I_{j+1,2}^{2}\xi^{j}+(1-\xi)^{2}I_{0,1}^{3}+(1-\xi)^{2}\sum_{j=0}^{\infty}I_{j+1,2}^{3}\xi^{j},
\end{split}
\end{equation}
Substituting the relations $
I_{n,2}^{2}=I_{n,1}^{2}-I_{n}$ and $
I_{n,2}^{3}=I_{n,1}^{3}-I_{n,1}^{2}$ into \eqref{eq:phi32} deduces that
\begin{equation*}
\begin{split}
\varphi^{(3,2)}(\xi)
=&I(\xi)+(1-\xi)I_{1}^{2}(\xi)+(1-\xi^{2})\sum_{j=0}^{\infty}
I_{j+1,1}^{3}\xi^{j} \\
&-2(1-\xi)I_{1}^{3}(\xi)+(3-\xi)(1-\xi)I_{0,1}^{3} \\
=&(\frac{5}{6}+\frac{1}{6}\xi)I(\xi)+(1-\xi)(I_{1}^{2}(\xi)-2I_{1}^{3}(\xi)+\frac{1}{6}I(\xi))
\\
&+(1-\xi^{2})\sum_{j=0}^{\infty}I_{j+1,1}^{3}\xi^{j}+(3-\xi)(1-\xi)I_{0,1}^{3}.
\end{split}
\end{equation*}
Based on Lemma \ref{le:1.5}, one hence obtains that
\begin{equation*}
I_{n,1}^{2}-2I_{n,1}^{3}+\frac{1}{6}I_{n}=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}s(1-s)\mathrm{d}s, \hspace{0.5cm}n\ge 0
\end{equation*}
is a completely monotonic sequence. Therefore the sequence can be expressed by
\begin{equation}\label{eq:lnmix32}
I_{n,1}^{2}-2I_{n,1}^{3}+\frac{1}{6}I_{n}=\int_{0}^{1}r^{n}\mathrm{d}\mu(r),\hspace{0.5cm} n=0,1,\cdots
\end{equation}
together with the function $\mu(r)$ being non-decreasing on interval $[0, 1]$. Therefore, from formulae \eqref{eq:ln}, \eqref{eq:lnmix32} and \eqref{eq:ln13}, it follows
\begin{equation*}
\begin{split}
\mathrm{Re}(\varphi^{(3,2)}(\xi))=&\int_{0}^{1}\mathrm{Re}(\frac{\frac{5}{6}+\frac{1}{6}\xi}{1-r\xi})\mathrm{d}\sigma(r)
+\int_{0}^{1}\mathrm{Re}(\frac{1-\xi}{1-r\xi})\mathrm{d}\mu(r) \\
+&\int_{0}^{1}r\mathrm{Re}(\frac{1-\xi^{2}}{1-r\xi})\mathrm{d}\beta(r)+\mathrm{Re}((3-\xi)(1-\xi))I_{0,1}^{3} \\
\ge& \frac{2}{3}\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\sigma(r)\ge \frac{I_{0}}{3}.
\end{split}
\end{equation*}
In the case of $(k,i)=(3,3)$, the definition of series $\omega^{(3,3)}(\xi)$ coincides with
\begin{equation}
\begin{split}
\omega^{(3,3)}(\xi)&=(1-\xi)(I_{0}+I_{1})+(1-\xi)\sum_{j=0}^{\infty}I_{j+2}\xi^{j}+(1-\xi)^{2}(I_{0,1}^{2}+I_{1,2}^{2}) \\
&+(1-\xi)^{2}\sum_{j=0}^{\infty}I_{j+2,3}^{2}\xi^{j}+(1-\xi)^{3}(I_{0,1}^{3}+I_{1,2}^{3})+(1-\xi)^{3}\sum_{j=0}^{\infty}I_{j+2,3}^{3}\xi^{j},
\end{split}
\end{equation}
therefore it follows that
\begin{equation}
\label{eq:phi33}
\begin{split}
\varphi^{(3,3)}(\xi)=&I_{0}+I_{1}+\sum_{j=0}^{\infty}I_{j+2}\xi^{j}+(1-\xi)\left(I_{0,1}^{2}+I_{1,2}^{2}\right)+(1-\xi)\sum_{j=0}^{\infty}I_{j+2,3}^{2}\xi^{j} \\
&+(1-\xi)^{2}\left(I_{0,1}^{3}+I_{1,2}^{3}\right)+(1-\xi)^{2}\sum_{j=0}^{\infty}I_{j+2,3}^{3}\xi^{j}. \\
\end{split}
\end{equation}
In addition, substituting the relations
\begin{equation*}
\begin{split}
&I_{n,3}^{2}=I_{n,2}^{2}-I_{n}=I_{n,1}^{2}-2I_{n}, \\
&I_{n,3}^{3}=I_{n,2}^{3}-I_{n,2}^{2}=I_{n,1}^{3}-2I_{n,1}^{2}+I_{n}, \hspace{0.5cm} n\ge 0
\end{split}
\end{equation*}
into \eqref{eq:phi33} yields that
\begin{equation*}
\begin{split}
\varphi^{(3,3)}(\xi)
=&I_{0}+I_{1}+\sum_{j=0}^{\infty}I_{j+2}\xi^{j}+(1-\xi)\left(I_{0,1}^{2}+I_{1,1}^{2}-I_{1}\right)+(1-\xi)\sum_{j=0}^{\infty}\left(I_{j+2,1}^{2}-2I_{j+2}\right)\xi^{j} \\
&+(1-\xi)^{2}\left(I_{0,1}^{3}+I_{1,
1}^{3}-I_{1,1}^{2}\right)+(1-\xi)^{2}\sum_{j=0}^{\infty}\left(I_{j+2,1}^{3}-2I_{j+2,1}^{2}+I_{j+2}\right)\xi^{j} \\
=&I(\xi)+(1-\xi)I_{1}^{2}(\xi)-2(1-\xi)I_{1}^{3}(\xi)+(1-\xi^{2})\sum_{j=0}^{\infty}I_{j+2,2}^{3}\xi^{j}-(3-4\xi+\xi^{2})\sum_{j=0}^{\infty}I_{j+1,2}^{3}\xi^{j} \\
&+(3-\xi)(1-\xi)\left(I_{0,1}^{3}+I_{1,2}^{3}\right)+\left(1-\xi^{2}\right)\sum_{j=0}^{\infty}I_{j+1,1}^{3}\xi^{j}.
\end{split}
\end{equation*}
In view of Lemma \ref{le:Ijqr}, sequence $(-I_{n,2}^{3})_{n=0}^{\infty}$ is completely monotonic, thus there exists a non-decreasing function $\vartheta(r)$ on $[0, 1]$ such that
\begin{equation*}
-I_{n,2}^{3}=\int_{0}^{1}r^{n}\mathrm{d}\vartheta(r), \hspace{0.5cm} n=0, 1, \cdots,
\end{equation*}
which yields
\begin{equation*}
(1-\xi^{2})\sum_{j=0}^{\infty}I_{j+2,2}^{3}\xi^{j}-(3-4\xi+\xi^{2})\sum_{j=0}^{\infty}I_{j+1,2}^{3}\xi^{j}
=\int_{0}^{1}\frac{(3r-r^{2})-4r\xi+(r^{2}+r)\xi^{2}}{1-r\xi}\mathrm{d}\vartheta(r)
\end{equation*}
for $|\xi|<1$. It follows
\begin{equation*}
\begin{split}
&\mathrm{Re}\left(\frac{(3r-r^{2})-4r\xi+(r^{2}+r)\xi^{2}}{1-r\xi}\right) \\
=&\frac{(3r-r^{2})(1-r|\xi|\cos\theta)+4r^{2}|\xi|^{2}-4r|\xi|\cos\theta+(r^{2}+r)|\xi|^{2}\cos2\theta-(r^{3}+r^{2})|\xi|^{3}\cos\theta}{1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}} \\
:=&f(r,|\xi|,\theta).
\end{split}
\end{equation*}
A calculation yields that
\begin{equation}\label{eq:pf}
\begin{split}
\frac{\partial f}{\partial \theta}(r,|\xi|,\theta)
&=\frac{r|\xi|\sin\theta}{(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2})^{2}}g(r,|\xi|,\theta),
\end{split}
\end{equation}
where
\begin{equation*}
\begin{split}
g(r,|\xi|,\theta)=&\left(4+3r-r^{2}-4(r+1)|\xi|\cos\theta+(r^{2}+r)|\xi|^{2}\right)(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}) \\
-&2\left((3r-r^{2})(1-r|\xi|\cos\theta)+4r^{2}|\xi|^{2}-4r|\xi|\cos\theta+(r^{2}+r)|\xi|^{2}\cos2\theta-(r^{3}+r^{2})|\xi|^{3}\cos\theta\right),
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\frac{\partial g}{\partial \theta}(r,|\xi|,\theta)=4|\xi|\sin\theta(r+1)(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}).
\end{split}
\end{equation*}
We thus know that $g(r,|\xi|,0)\le g(r,|\xi|,\theta)\le g(r,|\xi|,\pi)$. In addition, there is
\begin{equation*}
g(r,|\xi|,0)=(4-3r+r^{2})-4(1+r)|\xi|+(7r+3r^{2}+3r^{3}-r^{4})|\xi|^{2}-4(r^{3}+r^{2})|\xi|^{3}+(r^{3}+r^{4})|\xi|^{4},
\end{equation*}
and
\begin{equation}\label{eq:pg}
\frac{\partial g}{\partial |\xi|}(r,|\xi|,0)=-4(1+r)+2(7r+3r^{2}+3r^{3}-r^{4})|\xi|-12(r^{3}+r^{2})|\xi|^{2}+4(r^{3}+r^{4})|\xi|^{3}.
\end{equation}
In view of
\begin{equation*}
\frac{\partial^{2} g}{\partial |\xi|^{2}}(r,|\xi|,0)=r\left(12(r+1)(r|\xi|-1)^{2}+2(1-r)^{3}\right)\ge 0
\end{equation*}
for all $0\le r\le 1$ and $0\le|\xi|<1$, we can therefore obtain that $\frac{\partial g}{\partial |\xi|}(r,|\xi|,0)< \frac{\partial g}{\partial |\xi|}(r,1,0)$. In addition, formula \eqref{eq:pg} shows that
\begin{equation*}
\frac{\partial g}{\partial |\xi|}(r,1,0)=2(r^{3}-3r+2)(r-1)\le 0
\end{equation*}
for all $0\le r\le 1$, it derives that $\frac{\partial g}{\partial |\xi|}(r,|\xi|,0)<\frac{\partial g}{\partial |\xi|}(r,1,0)\le 0$ for all $0\le r\le 1$ and $0\le|\xi|<1$. We can finally get that
\begin{equation*}
g(r,|\xi|,0)>g(r,1,0)=0,\hspace{0.5cm} 0\le r\le 1, \hspace{0.2cm} 0\le|\xi|<1.
\end{equation*}
Hence, it holds that $g(r, |\xi|, \theta)\ge g(r,|\xi|,0)>0$ in the cases of $0\le r\le 1$ and $0\le |\xi|<1$. According to formula \eqref{eq:pf}, we have that $f(r, |\xi|, 0)\le f(r, |\xi|, \theta)\le f(r, |\xi|, \pi)$ for all $0\le r\le 1$ and $0\le |\xi|<1$. The definition of $f(r,|\xi|,\theta)$ states
\begin{equation*}
f(r,|\xi|,0)=\frac{3r-r^{2}-4r|\xi|+r^{2}|\xi|^{2}+r|\xi|^{2}}{1-r|\xi|}.
\end{equation*}
Taking the derivative with respect to $|\xi|$ obtains
\begin{equation*}
\frac{\partial f}{\partial |\xi|}(r,|\xi|,0)=\frac{r}{(1-r|\xi|)^{2}}\left(-4+3r-r^{2}+2(r+1)|\xi|-(r^{2}+r)|\xi|^{2}\right)=\frac{r}{(1-r|\xi|)^{2}}h(r,|\xi|).
\end{equation*}
It can be easily checked that $\frac{\partial h}{\partial |\xi|}(r,|\xi|)\ge 2(1-r^{2})\ge 0$ for $0\le r\le 1$, in combination with $h(r,1)=-2(1-r)^{2}\le 0$, we have that $h(r,|\xi|)\le h(r,1)\le 0$ for $0\le r\le 1$. And consequently, result $\frac{\partial f}{\partial |\xi|}(r,|\xi|,0)\le 0$ suggests that $f(r,|\xi|,0)\ge f(r,1,0)$ for
$0\le r\le 1$ and $0\le |\xi|<1$. In combination with $f(r,1,0)=0$, we can obtain that
\begin{equation*}
f(r,|\xi|,\theta)\ge f(r,|\xi|,0)\ge 0, \hspace{0.5cm}\forall~ 0\le r\le 1, \hspace{0.2cm} |\xi|<1,\hspace{0.2cm} \theta\in\mathbb{R}.
\end{equation*}
Therefore, it follows | 3,970 | 54,748 | en |
train | 0.45.11 | &=\frac{r|\xi|\sin\theta}{(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2})^{2}}g(r,|\xi|,\theta),
\end{split}
\end{equation}
where
\begin{equation*}
\begin{split}
g(r,|\xi|,\theta)=&\left(4+3r-r^{2}-4(r+1)|\xi|\cos\theta+(r^{2}+r)|\xi|^{2}\right)(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}) \\
-&2\left((3r-r^{2})(1-r|\xi|\cos\theta)+4r^{2}|\xi|^{2}-4r|\xi|\cos\theta+(r^{2}+r)|\xi|^{2}\cos2\theta-(r^{3}+r^{2})|\xi|^{3}\cos\theta\right),
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\frac{\partial g}{\partial \theta}(r,|\xi|,\theta)=4|\xi|\sin\theta(r+1)(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}).
\end{split}
\end{equation*}
We thus know that $g(r,|\xi|,0)\le g(r,|\xi|,\theta)\le g(r,|\xi|,\pi)$. In addition, there is
\begin{equation*}
g(r,|\xi|,0)=(4-3r+r^{2})-4(1+r)|\xi|+(7r+3r^{2}+3r^{3}-r^{4})|\xi|^{2}-4(r^{3}+r^{2})|\xi|^{3}+(r^{3}+r^{4})|\xi|^{4},
\end{equation*}
and
\begin{equation}\label{eq:pg}
\frac{\partial g}{\partial |\xi|}(r,|\xi|,0)=-4(1+r)+2(7r+3r^{2}+3r^{3}-r^{4})|\xi|-12(r^{3}+r^{2})|\xi|^{2}+4(r^{3}+r^{4})|\xi|^{3}.
\end{equation}
In view of
\begin{equation*}
\frac{\partial^{2} g}{\partial |\xi|^{2}}(r,|\xi|,0)=r\left(12(r+1)(r|\xi|-1)^{2}+2(1-r)^{3}\right)\ge 0
\end{equation*}
for all $0\le r\le 1$ and $0\le|\xi|<1$, we can therefore obtain that $\frac{\partial g}{\partial |\xi|}(r,|\xi|,0)< \frac{\partial g}{\partial |\xi|}(r,1,0)$. In addition, formula \eqref{eq:pg} shows that
\begin{equation*}
\frac{\partial g}{\partial |\xi|}(r,1,0)=2(r^{3}-3r+2)(r-1)\le 0
\end{equation*}
for all $0\le r\le 1$, it derives that $\frac{\partial g}{\partial |\xi|}(r,|\xi|,0)<\frac{\partial g}{\partial |\xi|}(r,1,0)\le 0$ for all $0\le r\le 1$ and $0\le|\xi|<1$. We can finally get that
\begin{equation*}
g(r,|\xi|,0)>g(r,1,0)=0,\hspace{0.5cm} 0\le r\le 1, \hspace{0.2cm} 0\le|\xi|<1.
\end{equation*}
Hence, it holds that $g(r, |\xi|, \theta)\ge g(r,|\xi|,0)>0$ in the cases of $0\le r\le 1$ and $0\le |\xi|<1$. According to formula \eqref{eq:pf}, we have that $f(r, |\xi|, 0)\le f(r, |\xi|, \theta)\le f(r, |\xi|, \pi)$ for all $0\le r\le 1$ and $0\le |\xi|<1$. The definition of $f(r,|\xi|,\theta)$ states
\begin{equation*}
f(r,|\xi|,0)=\frac{3r-r^{2}-4r|\xi|+r^{2}|\xi|^{2}+r|\xi|^{2}}{1-r|\xi|}.
\end{equation*}
Taking the derivative with respect to $|\xi|$ obtains
\begin{equation*}
\frac{\partial f}{\partial |\xi|}(r,|\xi|,0)=\frac{r}{(1-r|\xi|)^{2}}\left(-4+3r-r^{2}+2(r+1)|\xi|-(r^{2}+r)|\xi|^{2}\right)=\frac{r}{(1-r|\xi|)^{2}}h(r,|\xi|).
\end{equation*}
It can be easily checked that $\frac{\partial h}{\partial |\xi|}(r,|\xi|)\ge 2(1-r^{2})\ge 0$ for $0\le r\le 1$, in combination with $h(r,1)=-2(1-r)^{2}\le 0$, we have that $h(r,|\xi|)\le h(r,1)\le 0$ for $0\le r\le 1$. And consequently, result $\frac{\partial f}{\partial |\xi|}(r,|\xi|,0)\le 0$ suggests that $f(r,|\xi|,0)\ge f(r,1,0)$ for
$0\le r\le 1$ and $0\le |\xi|<1$. In combination with $f(r,1,0)=0$, we can obtain that
\begin{equation*}
f(r,|\xi|,\theta)\ge f(r,|\xi|,0)\ge 0, \hspace{0.5cm}\forall~ 0\le r\le 1, \hspace{0.2cm} |\xi|<1,\hspace{0.2cm} \theta\in\mathbb{R}.
\end{equation*}
Therefore, it follows
\begin{equation*}
\begin{split}
\mathrm{Re}(\varphi^{(3,3)}(\xi))=&\int_{0}^{1}\mathrm{Re}(\frac{\frac{5}{6}+\frac{1}{6}\xi}{1-r\xi})\mathrm{d}\sigma(r)
+\int_{0}^{1}\mathrm{Re}(\frac{1-\xi}{1-r\xi})\mathrm{d}\mu(r)+\int_{0}^{1}r\mathrm{Re}(\frac{1-\xi^{2}}{1-r\xi})\mathrm{d}\beta(r) \\
+&\int_{0}^{1}\mathrm{Re}\left(\frac{(3r-r^{2})-4r\xi+(r^{2}+r)\xi^{2}}{1-r\xi}\right)\mathrm{d}\vartheta(r)+\mathrm{Re}\left((3-\xi)(1-\xi)\right)(I_{0,1}^{3}
+I_{1,2}^{3}) \\
\ge& \frac{2}{3}\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\sigma(r)\ge \frac{I_{0}}{3},
\end{split}
\end{equation*}
since there holds that $I_{0,1}^{3}+I_{1,2}^{3}=\frac{2^{1-\alpha}(\alpha^{2}+\alpha)}{3\Gamma(4-\alpha)}\ge0$ for all $0\le \alpha\le 1$.
In addition, for $\xi=1$, assume that $\varphi^{(k,i)}(1)=0$, we know from the definition of $\varphi^{(k,i)}(\xi)$ that
\begin{equation}
\label{eq:id1.39}
\varphi^{(k,i)}(\xi)=I(\xi)+l^{(k,i)}(\xi),
\end{equation}
where the coefficients of series $l^{(k,i)}(\xi)$ is absolutely convergent. The definition of coefficients of $I(\xi)$ yields that $\sum_{i=0}^{n}I_{i}$ is arbitrary large as increasing $n$. However, the boundedness of $l^{(k,i)}(1)$ contradicts identity \eqref{eq:id1.39} for $\xi=1$, which obtains $\varphi^{(k,i)}(1)\ne 0$.
In the rest case of $|\xi|=1$ and $\xi\ne 1$, we can get from Corollary \ref{co:1} that
series $\varphi^{(k,i)}(\xi)$ is pointwise continuous on $|\xi|\le 1$ except $\xi=1$, then for the sequence $\xi_{n}=(1-\frac{1}{n})\xi$ satisfying $|\xi_{n}|<1$ for all $n\in \mathbb{N}^{+}$, $\varphi^{(k,i)}(\xi)$ is the limit point of sequence $\varphi^{(k,i)}(\xi_{n})$, thus
\begin{equation*}
\mathrm{Re}( \varphi^{(k,i)}(\xi))=\lim_{n\to +\infty}\mathrm{Re}(\varphi^{(k,i)}(\xi_{n}))\ge c^{(k,i)}>0,
\end{equation*}
where constants $c^{(k,i)}$ are independent of $n$.
\end{proof} | 2,297 | 54,748 | en |
train | 0.45.12 | \section{Convergence analysis}\label{conversection}
In this section, we consider the global error estimation for the problem \eqref{eq:nolinfode} when the numerical approximations \eqref{eq:nnonliode} are employed. Assume that $u(t_{n})$ is the exact solution of \eqref{eq:nolinfode} at $t=t_{n}$, then it satisfies
\begin{equation}\label{eq:exac}
D_{k,i}^{\alpha}u(t_{n})=f(t_{n}, u(t_{n}))+\tau_{n}^{(k,i)},\hspace{0.618cm} k\le n\le N,
\end{equation}
where the difference operator $D_{k,i}^{\alpha}$ is defined by \eqref{Dki1} and the local truncation error $\tau_{n}^{(k,i)}$ is denoted by \eqref{eq:trunc}. Suppose that $u_{n}^{(k,i)}$ is the solution of \eqref{eq:nnonliode} for each pair of $(k,i)$, we denote the global error by
\begin{equation}
e_{n}^{(k,i)}=u(t_{n})-u_{n}^{(k,i)} \hspace{0.618cm}\text{for} \hspace{0.2cm}0\le n\le N,
\end{equation}
where $e_{0}^{(k,i)}=0$. Thus subtracting \eqref{eq:nnonliode} by \eqref{eq:exac} implies that
\begin{equation}\label{eq:Dkie}
D_{k,i}^{\alpha}e_{n}^{(k,i)}=\delta f_{n}^{(k,i)}+\tau_{n}^{(k,i)},\hspace{0.618cm} k\le n\le N,
\end{equation}
where the notation $\delta f_{n}^{(k,i)}$ is denoted by $f(t_{n}, u(t_{n}))-f(t_{n}, u_{n}^{(k,i)})$. In addition, substituting \eqref{Dki1} into \eqref{eq:Dkie} yields
\begin{equation}\label{eq:node}
\sum_{m=0}^{k-1}w_{n,m}^{(k,i)}e_{m}^{(k,i)}+\sum_{j=0}^{n}\omega_{n-j}^{(k,i)}e_{j}^{(k,i)}
=(\Delta t)^{\alpha}\delta f_{n}^{(k,i)}+(\Delta t)^{\alpha}\tau_{n}^{(k,i)},\hspace{0.618cm} k\le n\le N.
\end{equation}
Multiplying $\xi^{n-k}$ on both sides of \eqref{eq:node} and summing up for all $n\ge k$, one obtains
\begin{equation*}
\begin{split}
\sum_{n=0}^{\infty}\sum_{m=0}^{k-1}&\left(w_{n+k,m}^{(k,i)}+\omega_{n+k-m}^{(k,i)}\right)e_{m}^{(k,i)}\xi^{n}+\sum_{n=0}^{\infty}\sum_{j=k}^{n+k}\omega_{n+k-j}^{(k,i)}e_{j}^{(k,i)}\xi^{n}\\
&=(\Delta t)^{\alpha}\sum_{n=0}^{\infty}\delta f_{n+k}^{(k,i)}\xi^{n}+(\Delta t)^{\alpha}\sum_{n=0}^{\infty}\tau_{n+k}^{(k,i)}\xi^{n},
\end{split}
\end{equation*}
since
\begin{equation*}
\sum_{n=0}^{\infty}\sum_{j=k}^{n+k}\omega_{n+k-j}^{(k,i)}e_{j}^{(k,i)}\xi^{n}=
\sum_{n=0}^{\infty}\sum_{j=0}^{n}\omega_{n-j}^{(k,i)}e_{j+k}^{(k,i)}\xi^{n}
=\sum_{j=0}^{\infty}e_{j+k}^{(k,i)}\xi^{j}\sum_{n=0}^{\infty}\omega_{n}^{(k,i)}\xi^{n},
\end{equation*}
it follows
\begin{equation}\label{eq:omegakie}
\omega^{(k,i)}(\xi)e^{(k,i)}(\xi)
=\sum_{m=0}^{k-1}e_{m}^{(k,i)}s_{m}^{(k,i)}(\xi)+(\Delta t)^{\alpha}\delta f^{(k,i)}(\xi)+(\Delta t)^{\alpha}\tau^{(k,i)}(\xi),
\end{equation}
where
\begin{equation}\label{eq:denote1}
\begin{split}
&s_{m}^{(k,i)}(\xi):=\sum_{n=0}^{\infty}s_{n,m}^{(k,i)}\xi^{n}=-\sum_{n=0}^{\infty}\left(w_{n+k,m}^{(k,i)}+\omega_{n+k-m}^{(k,i)}\right)\xi^{n},\hspace{0.618cm} e^{(k,i)}(\xi)=\sum_{n=0}^{\infty}e_{n+k}^{(k,i)}\xi^{n}, \\
&\omega^{(k,i)}(\xi)=\sum_{n=0}^{\infty}\omega_{n}^{(k,i)}\xi^{n},\hspace{0.718cm}
\delta f^{(k,i)}(\xi)=\sum_{n=0}^{\infty}\delta f_{n+k}^{(k,i)}\xi^{n},\hspace{0.618cm}
\tau^{(k,i)}(\xi)=\sum_{n=0}^{\infty}\tau_{n+k}^{(k,i)}\xi^{n}.
\end{split}
\end{equation}
\begin{lemma}
For $1\le i\le k\le3$ and $0\le m\le k-1$, the coefficients $s_{n,m}^{(k,i)}$ are denoted by \eqref{eq:denote1}. Then for all $n \ge 0$, it holds that $s_{n,m}^{(k,i)}$ is bounded, and there exist some bounded constants $c_{m}^{(k,i)}>0$ , which are independent of $n$ and $\alpha$, such that
\begin{equation}\label{eq:ski}
|s_{n,0}^{(k,i)}|\le \frac{c_{0}^{(k,i)}n^{-\alpha}}{\Gamma(1-\alpha)},\hspace{0.9cm}
|s_{n,m}^{(k,i)}|\le \frac{c_{m}^{(k,i)}n^{-\alpha-1}}{|\Gamma(-\alpha)|}
\end{equation}
for $n\ge 1$ and $m\ge 1$.
\end{lemma}
\begin{proof}
It is known from \eqref{ljqr} that for any finite $q, r\in\mathbb{N}^{+}$, $I_{n,q}^{r}$ is bounded for all $n\in\mathbb{Z}$. Since the coefficients $s_{n,m}^{(k,i)}$ are denoted as the linear combinations of $I_{n,q}^{r}$, we can immediately obtain the boundedness of $s_{n,m}^{(k,i)}$ for all integer $n\ge 0$.
Moreover, in the cases of $1\le i\le k\le 3$, each $s_{n,0}^{(k,i)}$ can be expressed as a linear combination of $I_{l}$ and $I_{l,1}^{r}$ with $l\ge n$ and $1\le r\le 3$. Based on formulae \eqref{eq:inqr} and \eqref{eq:inqr1}, it yields $I_{n}=O\left(\frac{n^{-\alpha}}{\Gamma(1-\alpha)}\right)$ and $I_{n,1}^{r}=O\left(\frac{n^{-\alpha-1}}{\Gamma(-\alpha)}\right)=o\left(\frac{n^{-\alpha}}{\Gamma(1-\alpha)}\right)$ for $r\ge 2$ and $n\ge 1$, respectively. Therefore, it implies that there is a uniform bound with respect to $n$ and $\alpha$, denoted by $c_{0}^{(k,i)}>0$, such that $|s_{n,0}^{(k,i)}|\le \frac{c_{0}^{(k,i)}n^{-\alpha}}{\Gamma(1-\alpha)}$ as $n\ge 1$.
In terms of $m\ge 1$, observe that $s_{n,m}^{(k,i)}$ are the linear combinations of $\nabla I_{l}$, $I_{l,1}^{r}$ and $\nabla^{p}I_{l,1}^{r}$, for $l\ge n+1$, $r\ge 2$ and $1\le p\le 3$. According to formulae \eqref{eq:nablainqr} and \eqref{eq:nablakinqr1}, we know that
$\nabla I_{n}=O\left(\frac{(n-1)^{-\alpha-1}}{\Gamma(-\alpha)}\right)=O\left(\frac{n^{-\alpha-1}}{\Gamma(-\alpha)}\right)$ and
$\nabla^{p}I_{n,1}^{r}=O\left(\frac{(n-p)^{-\alpha-p-1}}{\Gamma(-\alpha-p+1)}\right)=o\left(\frac{n^{-\alpha-1}}{\Gamma(-\alpha)}\right)$, therefore it holds $s_{n,m}^{(k,i)}=O\left(\frac{n^{-\alpha-1}}{\Gamma(-\alpha)}\right)$, and hence there exist constants $c_{m}^{(k,i)}>0$ such that the last inequality of \eqref{eq:ski} is satisfied.
\end{proof} | 2,300 | 54,748 | en |
train | 0.45.13 | According to the definition of the series $\omega^{(k,i)}(\xi)$, it is important to notice the decompositions of the form
\begin{equation}\label{eq:rela}
\omega^{(k,i)}(\xi)=(1-\xi)\varphi^{(k,i)}(\xi)=(1-\xi)^{\alpha}\psi^{(k,i)}(\xi),\hspace{0.5cm}\text{for}~~0<\alpha<1,
\end{equation}
where series $\varphi^{(k,i)}(\xi)$ is defined by \eqref{eq:fomega} and denote that
\begin{equation}\label{eq:psiki}
\psi^{(k,i)}(\xi)=(1-\xi)^{1-\alpha}\varphi^{(k,i)}(\xi).
\end{equation}
Formula \eqref{eq:rela} indicates a relationship between the proposed method and the fractional Euler method mentioned in \cite{LubichC:1986a}.
In the following part, we would like to discuss some relevant properties of the series $\psi^{(k,i)}(\xi)$ as preliminaries.
\begin{lemma}\label{le:gn}
Assume that sequences $\{g_{n}^{(\beta)}\}_{n=0}^{\infty}$ are generated by the power series $(1-\xi)^{\beta}$ for $\beta\in\mathbb{R}$, i.e.,
\begin{equation}\label{eq:gn}
(1-\xi)^{\beta}=\sum_{n=0}^{\infty}(-1)^{n}\binom{\beta}{n}\xi^{n}=\sum_{n=0}^{\infty}g_{n}^{(\beta)}\xi^{n}.
\end{equation}
Therefore, in the cases of $\beta\in(-1,1)$, based on \eqref{eq:gn}, there holds
\begin{equation}\label{eq:gn1}
\left\{
\begin{split}
\beta\in (-1,0): &\hspace{0.2cm}g_{0}^{(\beta)}=1,\hspace{0.4cm}g_{0}^{(\beta)}>g_{1}^{(\beta)}>\cdots>0,\\
&\hspace{0.2cm}\sum_{i=0}^{n}g_{i}^{(\beta)}=g_{n}^{(\beta-1)}, \hspace{0.2cm} n\ge 0; \\
\beta\in (0,1):\hspace{0.3cm} &\hspace{0.2cm}g_{0}^{(\beta)}=1,\hspace{0.4cm}g_{n}^{(\beta)}<0,\hspace{0.2cm} n\ge 1, \\
&\hspace{0.2cm}1>|g_{1}^{(\beta)}|>|g_{2}^{(\beta)}|>\cdots>0, \\
&\hspace{0.2cm}\sum_{i=0}^{\infty}g_{i}^{(\beta)}=0,\hspace{0.4cm}\sum_{i=0}^{n}g_{i}^{(\beta)}=g_{n}^{(\beta-1)}, \hspace{0.2cm} n\ge 0. \\
\end{split}
\right.
\end{equation}
\end{lemma}
\begin{lemma}\label{le:le1}
In the cases of $1\le i\le k\le 6$, the coefficients of the power series $\psi^{(k,i)}(\xi)$ belong to $l_{1}$ space.
\end{lemma}
\begin{proof}
According to the expression of $\varphi^{(k,i)}(\xi)$ presented in Theorem \ref{coro:1}, there holds
\begin{equation}\label{rela:1}
\varphi^{(k,i)}(\xi)=I(\xi)+l^{(k,i)}(\xi), \hspace{0.9cm}\text{with}\hspace{0.3cm} \sum_{n=0}^{\infty}|l_{n}^{(k,i)}|<\infty.
\end{equation}
In addition, together with \eqref{eq:psiki}, it follows
\begin{equation*}
\psi^{(k,i)}(\xi)=(1-\xi)^{(1-\alpha)}I(\xi)+(1-\xi)^{(1-\alpha)}l^{(k,i)}(\xi),
\end{equation*}
therefore, it suffices to prove that the coefficients of series $(1-\xi)^{1-\alpha}I(\xi)$ belong to $l_{1}$ space.
From the gamma function's definition of the form
\begin{equation*}
\Gamma(\beta)=\lim_{n\to\infty}\frac{n^{\beta}}{(-1)^{n}\binom{-\beta}{n}(n+\beta)},\hspace{0.618cm} \beta\ne 0, -1, -2, \cdots,
\end{equation*}
one obtains the asymptotically equal relation
\begin{equation}\label{eq:gammaasym}
\frac{n^{\beta-1}}{\Gamma(\beta)}\cong(-1)^{n}\binom{-
\beta}{n},\hspace{0.618cm}\text{as}\hspace{0.209cm}n\to\infty,
\end{equation}
where the notation $\cong$ means the ratio $\left(n^{\beta-1}/\Gamma(\beta)\right)\big/(-1)^{n}\binom{-
\beta}{n}\to 1$ as $n\to\infty$.
Furthermore, it is known from \cite{Erdelyi:1953, LubichC:1986b} that
\begin{equation}\label{eq:asymptgamma}
(-1)^{n}\binom{-\beta}{n}=\frac{n^{\beta-1}}{\Gamma(\beta)}\left(1+O\left(\frac{\beta-1}{n}\right)\right).
\end{equation}
On the other hand, based on the definition of $I_{n}$, it yields that $
I_{n}\cong \frac{n^{-\alpha}}{\Gamma(1-\alpha)}$ as $n\to\infty$, furthermore,
\begin{equation}
\begin{split}
\sum_{n=1}^{\infty}\left|I_{n}-\frac{n^{-\alpha}}{\Gamma(1-\alpha)}\right|&=\frac{1}{\Gamma(1-\alpha)}\sum_{n=1}^{\infty}\int_{0}^{1}\left(n^{-\alpha}-(n+1-s)^{-\alpha}\right)\mathrm{d}s \\
&=\frac{\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}\int_{0}^{1-s}\sum_{n=1}^{\infty}(n+t)^{-\alpha-1}\mathrm{d}t\mathrm{d}s \\
&\le\frac{\alpha}{\Gamma(1-\alpha)}\sum_{n=1}^{\infty}n^{-\alpha-1} \\
&\le \frac{\alpha}{\Gamma(1-\alpha)}\left(1+\int_{1}^{\infty}x^{-\alpha-1}\mathrm{d}x\right)\\
&=\frac{\alpha+1}{\Gamma(1-\alpha)}<+\infty.
\end{split}
\end{equation}
Therefore, in combination with \eqref{eq:asymptgamma}, it holds that
\begin{equation}\label{rela:2}
I_{n}=g_{n}^{(\alpha-1)}+v_{n},\hspace{0.409cm}\text{with}\hspace{0.209cm}\sum_{n=0}^{\infty}|v_{n}|<\infty,
\end{equation}
hence one has
\begin{equation*}
(1-\xi)^{1-\alpha}I(\xi)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}g_{n-k}^{(1-\alpha)}I_{k}\right)\xi^{n}
\end{equation*}
with the relationship that
\begin{equation*}
\begin{split}
\sum_{n=0}^{\infty}|\sum_{k=0}^{n}g_{n-k}^{(1-\alpha)}I_{k}|
&= \sum_{n=0}^{\infty}|\sum_{k=0}^{n}g_{n-k}^{(1-\alpha)}\left(g_{k}^{(\alpha-1)}+v_{k}\right)|\\
&\le \sum_{n=0}^{\infty}|\sum_{k=0}^{n}g_{n-k}^{(1-\alpha)}g_{k}^{(\alpha-1)}|+\sum_{n=0}^{\infty}|\sum_{k=0}^{n}g_{n-k}^{(1-\alpha)}v_{k}| \\
&\le 1+\sum_{n=0}^{\infty}|g_{n}^{(1-\alpha)}|\sum_{k=0}^{\infty}|v_{k}|<\infty,
\end{split}
\end{equation*}
which yields the desired result.
\end{proof}
\begin{lemma}\label{le:le2}
In the cases of $1\le i\le k\le 3$, it holds that $\psi^{(k,i)}(\xi)\ne 0$ for any $|\xi|\le 1$.
\end{lemma}
\begin{proof}
In the provement process of the Theorem \ref{coro:1}, it implies that $\varphi^{(k,i)}(\xi)\ne 0$ for all $|\xi|\le 1$ and $1\le i\le k\le 3$. The corresponding complex numbers $(1-\xi)^{1-\alpha}$ for any prescribed $|\xi|\le 1$ are located within the sector $S_{\alpha}=\{z: \hspace{0.2cm}|\mathrm{arg}(z)|\le \frac{(1-\alpha)\pi}{2}\}$. In addition, notice that $(1-\xi)^{1-\alpha}=0$ if and only if $\xi=1$. Thus, it remains to confirm the
value of the series $(1-\xi)^{1-\alpha}\varphi^{(k,i)}(\xi)$ at $\xi=1$. In fact, according
to the formulae \eqref{rela:1} and \eqref{rela:2}, it follows
\begin{equation*}
\begin{split}
\sum_{n=0}^{\infty}\sum_{l=0}^{n}g_{n-l}^{(1-\alpha)}\varphi_{l}^{(k,i)}
&= \sum_{n=0}^{\infty}\sum_{l=0}^{n}g_{n-l}^{(1-\alpha)}\left(g_{l}^{(\alpha-1)}+v_{l}+l_{l}^{(k,i)}\right)\\
&= \sum_{n=0}^{\infty}\sum_{l=0}^{n}g_{n-l}^{(1-\alpha)}g_{l}^{(\alpha-1)}+\sum_{n=0}^{\infty}\sum_{l=0}^{n}g_{n-l}^{(1-\alpha)}\left(v_{l}+l_{l}^{(k,i)}\right) \\
&= 1+\sum_{n=0}^{\infty}g_{n}^{(1-\alpha)}\sum_{l=0}^{\infty}\left(v_{l}+l_{l}^{(k,i)}\right)=1,
\end{split}
\end{equation*}
where the last equality holds in view of the Lemma \ref{le:gn}.
\end{proof} | 2,624 | 54,748 | en |
train | 0.45.14 | Therefore, according to the statements from Theorem \ref{th:pw}, Lemma \ref{le:le1} and Lemma \ref{le:le2}, we can immediately obtain the following result.
\begin{proposition}\label{pro:1}
For $1\le i\le k\le 3$ and $0<\alpha<1$, let
\begin{equation}\label{eq:rnki}
\frac{1}{\psi^{(k,i)}(\xi)}=r^{(k,i)}(\xi)=\sum_{n=0}^{\infty}r_{n}^{(k,i)}\xi^{n},
\end{equation}
then there exist bounded positive constants, denoted by $M_{\alpha}^{(k,i)}$, such that $\sum\limits_{n=0}^{\infty}
|r_{n}^{(k,i)}|=M_{\alpha}^{(k,i)}$ holds for each $k, i$.
\end{proposition}
\begin{theorem}
Let $u(t)$ and $\{u_{n}\}_{n=k}^{N}$ be the solutions of equations \eqref{eq:nolinfode} and \eqref{eq:node}, respectively. The function $f(t, u(t))$ in \eqref{eq:nolinfode} is assumed to satisfy the Lipschitz continuous condition with respect to the second variable $u$, and chosen properly such that the solution of \eqref{eq:nolinfode} is sufficiently smooth. Then
\begin{description}
\item [i)] in the cases of $1\le k\le 3$, it holds
\begin{equation}
|e_{n}^{(k,k)}|\le C^{(k,k)}\left(\sum_{m=0}^{k-1}|e_{m}^{(k,k)}|+\left(\Delta t\right)^{k+1-\alpha}t_{n-1}^{\alpha}\right),\hspace{0.618cm} k\le n\le N
\end{equation}
\item [ii)]in the cases of $1\le i<k\le 3$, it holds
\begin{equation}
|e_{n}^{(k,i)}|\le C^{(k,i)}\left(\sum_{m=0}^{k-1}|e_{m}^{(k,i)}|+\left(\Delta t\right)^{k}+\left(\Delta t\right)^{k+1-\alpha}t_{n-1}^{\alpha}\right),\hspace{0.618cm} k\le n\le N
\end{equation}
\end{description}
for sufficiently small $\Delta t>0$,
where $N\Delta t=T$ is fixed and constant $C^{(k,i)}>0$ is independent of $N$ and $n$.
\end{theorem}
\begin{proof}
Substituting formula \eqref{eq:rela} into \eqref{eq:omegakie} and using \eqref{eq:rnki}, one has
\begin{equation}\label{eq:nonlinear11}
\begin{split}
e^{(k,i)}(\xi)=\frac{r^{(k,i)}(\xi)}{(1-\xi)^{\alpha}}\left(\sum_{m=0}^{k-1}e_{m}^{(k,i)}s_{m}^{(k,i)}(\xi)+(\Delta t)^{\alpha}\delta f^{(k,i)}(\xi)+(\Delta t)^{\alpha}\tau^{(k,i)}(\xi)\right),
\end{split}
\end{equation}
which can also be written into a matrix-vector form
\begin{equation}\label{eq:MD12}
\begin{split}
\left[\begin{array}{l} e_{k}^{(k,i)} \\ e_{k+1}^{(k,i)} \\ \vdots \\ e_{N}^{(k,i)}\end{array}\right]
&=\left[\begin{array}{llll}
r_{0}^{(k,i)} & & & \\
r_{1}^{(k,i)} & r_{0}^{(k,i)} & & \\
\vdots & \ddots & \ddots & \\
r_{N-k}^{(k,i)} & \cdots & r_{1}^{(k,i)} & r_{0}^{(k,i)}
\end{array}\right]
\left[\begin{array}{llll}
g_{0}^{(-\alpha)} & & & \\
g_{1}^{(-\alpha)} & g_{0}^{(-\alpha)} & & \\
\vdots & \ddots & \ddots & \\
g_{N-k}^{(-\alpha)} & \cdots & g_{1}^{(-\alpha)} & g_{0}^{(-\alpha)}
\end{array}\right] \\
&\left(e_{0}^{(k,i)}
\left[\begin{array}{l} s_{0,0}^{(k,i)} \\ s_{1,0}^{(k,i)} \\ \vdots \\ s_{N-k,0}^{(k,i)}\end{array}\right]+\cdots
+e_{k-1}^{(k,i)}\left[\begin{array}{l} s_{0,k-1}^{(k,i)} \\ s_{1,k-1}^{(k,i)} \\ \vdots \\ s_{N-k,k-1}^{(k,i)}\end{array}\right]
+(\Delta t)^{\alpha}\left[\begin{array}{l} \delta f_{k}^{(k,i)} \\ \delta f_{k+1}^{(k,i)} \\ \vdots \\ \delta f_{N}^{(k,i)}\end{array}\right]+(\Delta t)^{\alpha}\left[\begin{array}{l} \tau_{k}^{(k,i)} \\ \tau_{k+1}^{(k,i)} \\ \vdots \\ \tau_{N}^{(k,i)}\end{array}\right]\right)
\end{split}
\end{equation}
with arbitrary $N\in\mathbb{N}$. Therefore for any $k\le n\le N$, it holds that
\begin{equation}\label{eq:}
\begin{split}
e_{n}^{(k,i)}&=\sum_{m=0}^{k-1}e_{m}^{(k,i)}\sum_{j=0}^{n-k}r_{n-k-j}^{(k,i)}\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}s_{i,m}^{(k,i)}+(\Delta t)^{\alpha}\sum_{j=0}^{n-k}r_{n-k-j}^{(k,i)}\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}\delta f_{i+k}^{(k,i)} \\
&\hspace{0.918cm}+(\Delta t)^{\alpha}\sum_{j=0}^{n-k}r_{n-k-j}^{(k,i)}\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}\tau_{i+k}^{(k,i)},
\end{split}
\end{equation}
where the coefficients $\{g_{n}^{(-\alpha)}\}$ are provided in
Lemma \ref{le:gn}. Since it is assumed that function $f(t, u(t))$ satisfies the Lipschitz continuous condition, there exists constant $L^{(k,i)}>0$ such that $|\delta f_{n}^{(k,i)}|\le L^{(k,i)}|e_{n}^{(k,i)}|$ for $k\le n\le N$. It follows
\begin{equation}\label{eq:enki}
\begin{split}
|e_{n}^{(k,i)}|&\le \sum_{m=0}^{k-1}|e_{m}^{(k,i)}|\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}|s_{i,m}^{(k,i)}| \\
&+(\Delta t)^{\alpha}\sum_{j=0}^{n-k}g_{n-k-j}^{(-\alpha)}\sum_{i=0}^{j}|r_{j-i}^{(k,i)}|\left(L^{(k,i)}|e_{i+k}^{(k,i)}|+|\tau_{i+k}^{(k,i)}|\right). \\
\end{split}
\end{equation}
On one hand, based on the relations \eqref{eq:ski} and \eqref{rela:2}, there exist constant $\tilde{c}_{k,i}>0$, such that $
|s_{n,0}^{(k,i)}|\le c_{k,i}\frac{n^{-\alpha}}{\Gamma(1-\alpha)}\le \tilde{c}_{k,i}g_{n}^{(\alpha-1)}$, one hence obtains
\begin{equation}\label{eq:ski0}
\begin{split}
\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}|s_{i,0}^{(k,i)}|&\le \tilde{c}_{k,i}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}g_{i}^{(\alpha-1)} \\
&\le \tilde{c}_{k,i}\sum_{j=0}^{\infty}|r_{j}^{(k,i)}|=\tilde{c}_{k,i}M_{\alpha}^{(k,i)},
\end{split}
\end{equation}
where $\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}g_{i}^{(\alpha-1)}=1$ for any $j\ge 0$ in view of the equality $(1-\xi)^{-\alpha}(1-\xi)^{\alpha-1}=(1-\xi)^{-1}$. On the other hand, there exist constant $\tilde{c}_{m}^{(k,i)}>0$, such that $|s_{n,m}^{(k,i)}|\le c_{m}^{(k,i)}\frac{n^{-\alpha-1}}{|\Gamma(-\alpha)|}\le \tilde{c}_{m}^{(k,i)}|g_{n}^{(\alpha)}|$, and for $m\ge 1$, it yields
\begin{equation}\label{eq:skim}
\begin{split}
\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|s_{l,m}^{(k,i)}|&\le \tilde{c}_{m}^{(k,i)}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|g_{l}^{(\alpha)}| \\
&\le 2\tilde{c}_{m}^{(k,i)}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{j}^{(-\alpha)}, \\
\end{split}
\end{equation}
since it is known that $\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}g_{l}^{(\alpha)}=0$ for any $j\ge 1$
according to the equality $(1-\xi)^{-\alpha}(1-\xi)^{\alpha}=1$, in combination with Lemma \ref{le:gn}, there yields $\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|g_{l}^{(\alpha)}|=g_{j}^{(-\alpha)}g_{0}^{(\alpha)}-\sum_{l=1}^{j}g_{j-l}^{(-\alpha)}g_{l}^{(\alpha)}=2g_{j}^{(-\alpha)}$. In addition, it is known that sequences $\{r_{n}^{(k,i)}\}$ belong to $l^{1}$ space,
and $g_{n}^{(-\alpha)}\to 0$ as $n\to \infty$, therefore, we know that the sequences $\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{j}^{(-\alpha)}\to 0$ as $n\to \infty$. Then, at least, the sequences $\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|s_{l,m}^{(k,i)}|$ can be bounded by $2\tilde{c}_{m}^{(k,i)}M_{\alpha}^{(k,i)}$.
In the cases of $1\le k\le 3$, recall that $|\tau_{n}^{(k,k)}|\le C_{\alpha}^{(k)}\left(\Delta t\right)^{k+1-\alpha}$ uniformly for $n\ge k$ in Theorem \ref{th:errorestmat}, it follows from \eqref{eq:gammaasym} that
\begin{equation}\label{eq:taukk}
\begin{split}
(\Delta t)^{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,k)}|\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}|\tau_{i+k}^{(k,k)}|
&=(\Delta t)^{\alpha}\sum_{j=0}^{n-k}g_{n-k-j}^{(-\alpha)}\sum_{i=0}^{j}|r_{j-i}^{(k,k)}||\tau_{i+k}^{(k,k)}|\\
&\le\left(\Delta t\right)^{k+1}C_{\alpha}^{(k)}M_{\alpha}^{(k,k)}\sum_{j=0}^{n-k}g_{j}^{(-\alpha)} \\
&\le\left(\Delta t\right)^{k+1}C_{\alpha}^{(k)}M_{\alpha}^{(k,k)}\big(1+C\sum_{j=1}^{n-k}\frac{j^{\alpha-1}}{\Gamma(\alpha)}\big) \\
&\le\left(\Delta t\right)^{k+1}C_{\alpha}^{(k)}M_{\alpha}^{(k,k)}\big(1+
\frac{C}{\Gamma(\alpha)}\int_{0}^{n-k}t^{\alpha-1}\mathrm{d}t\big) \\
&\le\tilde{C}_{\alpha}^{(k,k)}\left((\Delta t)^{k+1}+(\Delta t)^{k+1-\alpha}t_{n-k}^{\alpha}\right).
\end{split}
\end{equation}
In the other case of $1\le i<k\le 3$, according to Theorem \ref{th:errorestmat}, there exists constant $C_{\alpha}^{(k,i)}>0$, such that
\begin{equation*}
|\tau_{n}^{(k,i)}|\le C_{\alpha}^{(k,i)}\left((\Delta t)^{k-\alpha}\frac{(n-k)^{-\alpha-1}}{|\Gamma(-\alpha)|}+\frac{(\Delta t)^{k+1-\alpha}}{\Gamma(1-\alpha)}\right),
\end{equation*}
if $n\ge k$,
together with \eqref{eq:gammaasym}, it follows
\begin{equation}\label{eq:tauki}
\begin{split}
(\Delta t)^{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|\tau_{l+k}^{(k,i)}|&\le C_{\alpha}^{(k,i)}\Big((\Delta t)^{k}\tilde{c}_{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|g_{l}^{(\alpha)}| \\
&\hspace{1.236cm}+(\Delta t)^{k+1}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{n-k}g_{l}^{(-\alpha)}\Big) \\
&\le C_{\alpha}^{(k,i)}\Big(2(\Delta t)^{k}\tilde{c}_{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{j}^{(-\alpha)} \\
&\hspace{1.236cm}+(\Delta t)^{k+1}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{n-k}^{(-\alpha-1)}\Big)\\
&\le \tilde{C}_{\alpha}^{(k,i)}\left((\Delta t)^{k}+(\Delta t)^{k+1-\alpha}t_{n-k}^{\alpha}\right).
\end{split}
\end{equation}
Therefore formula \eqref{eq:enki} becomes
\begin{equation*}
\begin{split} | 3,974 | 54,748 | en |
train | 0.45.15 | \end{split}
\end{equation}
since it is known that $\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}g_{l}^{(\alpha)}=0$ for any $j\ge 1$
according to the equality $(1-\xi)^{-\alpha}(1-\xi)^{\alpha}=1$, in combination with Lemma \ref{le:gn}, there yields $\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|g_{l}^{(\alpha)}|=g_{j}^{(-\alpha)}g_{0}^{(\alpha)}-\sum_{l=1}^{j}g_{j-l}^{(-\alpha)}g_{l}^{(\alpha)}=2g_{j}^{(-\alpha)}$. In addition, it is known that sequences $\{r_{n}^{(k,i)}\}$ belong to $l^{1}$ space,
and $g_{n}^{(-\alpha)}\to 0$ as $n\to \infty$, therefore, we know that the sequences $\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{j}^{(-\alpha)}\to 0$ as $n\to \infty$. Then, at least, the sequences $\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|s_{l,m}^{(k,i)}|$ can be bounded by $2\tilde{c}_{m}^{(k,i)}M_{\alpha}^{(k,i)}$.
In the cases of $1\le k\le 3$, recall that $|\tau_{n}^{(k,k)}|\le C_{\alpha}^{(k)}\left(\Delta t\right)^{k+1-\alpha}$ uniformly for $n\ge k$ in Theorem \ref{th:errorestmat}, it follows from \eqref{eq:gammaasym} that
\begin{equation}\label{eq:taukk}
\begin{split}
(\Delta t)^{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,k)}|\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}|\tau_{i+k}^{(k,k)}|
&=(\Delta t)^{\alpha}\sum_{j=0}^{n-k}g_{n-k-j}^{(-\alpha)}\sum_{i=0}^{j}|r_{j-i}^{(k,k)}||\tau_{i+k}^{(k,k)}|\\
&\le\left(\Delta t\right)^{k+1}C_{\alpha}^{(k)}M_{\alpha}^{(k,k)}\sum_{j=0}^{n-k}g_{j}^{(-\alpha)} \\
&\le\left(\Delta t\right)^{k+1}C_{\alpha}^{(k)}M_{\alpha}^{(k,k)}\big(1+C\sum_{j=1}^{n-k}\frac{j^{\alpha-1}}{\Gamma(\alpha)}\big) \\
&\le\left(\Delta t\right)^{k+1}C_{\alpha}^{(k)}M_{\alpha}^{(k,k)}\big(1+
\frac{C}{\Gamma(\alpha)}\int_{0}^{n-k}t^{\alpha-1}\mathrm{d}t\big) \\
&\le\tilde{C}_{\alpha}^{(k,k)}\left((\Delta t)^{k+1}+(\Delta t)^{k+1-\alpha}t_{n-k}^{\alpha}\right).
\end{split}
\end{equation}
In the other case of $1\le i<k\le 3$, according to Theorem \ref{th:errorestmat}, there exists constant $C_{\alpha}^{(k,i)}>0$, such that
\begin{equation*}
|\tau_{n}^{(k,i)}|\le C_{\alpha}^{(k,i)}\left((\Delta t)^{k-\alpha}\frac{(n-k)^{-\alpha-1}}{|\Gamma(-\alpha)|}+\frac{(\Delta t)^{k+1-\alpha}}{\Gamma(1-\alpha)}\right),
\end{equation*}
if $n\ge k$,
together with \eqref{eq:gammaasym}, it follows
\begin{equation}\label{eq:tauki}
\begin{split}
(\Delta t)^{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|\tau_{l+k}^{(k,i)}|&\le C_{\alpha}^{(k,i)}\Big((\Delta t)^{k}\tilde{c}_{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|g_{l}^{(\alpha)}| \\
&\hspace{1.236cm}+(\Delta t)^{k+1}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{n-k}g_{l}^{(-\alpha)}\Big) \\
&\le C_{\alpha}^{(k,i)}\Big(2(\Delta t)^{k}\tilde{c}_{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{j}^{(-\alpha)} \\
&\hspace{1.236cm}+(\Delta t)^{k+1}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{n-k}^{(-\alpha-1)}\Big)\\
&\le \tilde{C}_{\alpha}^{(k,i)}\left((\Delta t)^{k}+(\Delta t)^{k+1-\alpha}t_{n-k}^{\alpha}\right).
\end{split}
\end{equation}
Therefore formula \eqref{eq:enki} becomes
\begin{equation*}
\begin{split}
|e_{n}^{(k,i)}|&\le(\Delta t)^{\alpha}L^{(k,i)}\Big(\sum_{j=0}^{n-k-1}g_{n-k-j}^{(-\alpha)}\sum_{l=0}^{j}|r_{j-l}^{(k,i)}||e_{l+k}^{(k,i)}|+g_{0}^{(-\alpha)}\sum_{l=0}^{n-k-1}|r_{n-k-l}^{(k,i)}||e_{l+k}^{(k,i)}| \\
&\hspace{2.618cm}+g_{0}^{(-\alpha)}|r_{0}^{(k,i)}||e_{n}^{(k,i)}|\Big)+\delta_{n}^{(k,i)}, \hspace{0.618cm}n\ge k.
\end{split}
\end{equation*}
If the time step $\Delta t>0$ is chosen sufficiently small, there exists bounded constants $c_{k,i}^{*}$ such that $0<\frac{1}{1-(\Delta t)^{\alpha}L^{(k,i)}g_{0}^{(-\alpha)}|r_{0}^{(k,i)}|}\le c_{k,i}^{*}$, and it follows
\begin{equation}\label{eq:enki}
\left\{
\begin{split}
|e_{k}^{(k,i)}|\le&\tilde{\delta}_{k}^{(k,i)}, \\
|e_{n}^{(k,i)}|\le&\tilde{\delta}_{n}^{(k,i)}+(\Delta t)^{\alpha}c_{k,i}^{*}L^{(k,i)}\Big(\sum_{j=0}^{n-k-1}g_{n-k-j}^{(-\alpha)}\sum_{l=0}^{j}|r_{j-l}^{(k,i)}||e_{l+k}^{(k,i)}|\\
&\hspace{0.618cm}+g_{0}^{(-\alpha)}\sum_{l=0}^{n-k-1}|r_{n-k-l}^{(k,i)}||e_{l+k}^{(k,i)}|\Big), \hspace{0.618cm}n\ge k+1,
\end{split}
\right.
\end{equation}
where denote by $\tilde{\delta}_{n}^{(k,i)}=c_{k,i}^{*}\delta_{n}^{(k,i)}$, and in the cases of $1\le k\le 3$, one has from \eqref{eq:ski0}, \eqref{eq:skim} and \eqref{eq:taukk} that
\begin{equation*}
\delta_{n}^{(k,k)}=C_{\alpha}^{(k,k)}\left(\sum_{m=0}^{k-1}|e_{m}^{(k,k)}|+\left(\Delta t\right)^{k+1-\alpha}t_{n-1}^{\alpha}\right), \hspace{0.618cm}n\ge k,
\end{equation*}
and in the cases of $1\le i<k\le 3$, from formulae \eqref{eq:ski0}, \eqref{eq:skim} and \eqref{eq:tauki}, it yields
\begin{equation*}
\delta_{n}^{(k,i)}=C_{\alpha}^{(k,i)}\left(\sum_{m=0}^{k-1}|e_{m}^{(k,i)}|+\left(\Delta t\right)^{k}+\left(\Delta t\right)^{k+1-\alpha}t_{n-1}^{\alpha}\right), \hspace{0.618cm}n\ge k.
\end{equation*}
The constants satisfy $C_{\alpha}^{(k,i)}=\max\{\tilde{c}_{k,i}M_{\alpha}^{(k,i)}, 2\tilde{c}_{m}^{(k,i)}M_{\alpha}^{(k,i)}, \tilde{C}_{\alpha}^{(k,i)} \}$. Then assume that the non-negative sequence $\{p_{n}^{(k,i)}\}_{n\ge 0}$ satisfies
\begin{equation}\label{eq:pnki}
\left\{
\begin{split}
&p_{0}^{(k,i)}=\tilde{\delta}_{k}^{(k,i)}, \\
&p_{n}^{(k,i)}=\tilde{\delta}_{n+k}^{(k,i)}+\frac{(\Delta t)^{\alpha}\tilde{L}^{(k,i)}}{\Gamma(\alpha)}\sum_{j=0}^{n-1}(n-j)^{\alpha-1}p_{j}^{(k,i)}, \hspace{0.618cm}n\ge 1,
\end{split}
\right.
\end{equation}
where the coefficient $\tilde{L}^{(k,i)}$ is chosen such that
\begin{equation*}
\tilde{L}^{(k,i)}=\max\{c_{k,i}^{*}L^{(k,i)}M_{\alpha}^{(k,i)}\left(1+\Gamma(\alpha)g_{1}^{(-\alpha)}\right), c_{k,i}^{*}L^{(k,i)}M_{\alpha}^{(k,i)}g_{n}^{(-\alpha)}n^{1-\alpha}\Gamma(\alpha)\}.
\end{equation*}
Therefore, according to the weakly singular discrete Gronwall inequality shown in \cite{DixonM:1986}, the monotonic increasing property of sequence $\{\tilde{\delta}_{n}^{(k,i)}\}_{n\ge 0}$ yields the sequence $\{p_{n}^{(k,i)}\}_{n\ge 1}$ is monotonic increasing with respect to $n$ for each $1\le i\le k\le 3$, and
correspondingly, it follows
\begin{equation*}
p_{n}^{(k,i)}\le \tilde{\delta}_{n+k}^{(k,i)}E_{\alpha}\left(\tilde{L}^{(k,i)}(n\Delta t)^{\alpha}\right), \hspace{0.618cm} n\ge 1,
\end{equation*}
where $E_{\alpha}(\cdot)$ is denoted by the Mittag-Leffler function.
In addition, according to \eqref{eq:enki} and \eqref{eq:pnki}, an induction process yields that $|e_{n}^{(k,i)}|\le p_{n-k}^{(k,i)}$ as $n\ge k$, thus, one obtains that
\begin{equation*}
|e_{n}^{(k,i)}|\le \tilde{\delta}_{n}^{(k,i)}E_{\alpha}\left(\tilde{L}^{(k,i)}(n-k)^{\alpha}\Delta t^{\alpha}\right)\le \tilde{\delta}_{n}^{(k,i)}E_{\alpha}\left(\tilde{L}^{(k,i)}T^{\alpha}\right)
\end{equation*}
in the case of $k\le n\le N$.
\end{proof} | 2,951 | 54,748 | en |
train | 0.45.16 | \begin{remark}
The provided convergence order is uniform for all $n\ge k$, especially suitable for the step $t_{n}$ near the origin. On the other hand, for those $t_{n}$ away from origin, the convergence result can be better. For example, it can be observed numerically if the computed starting values satisfy $u_{m}=u(t_{m})+O((\Delta t)^{k})$ for $1\le m\le k-1$, there holds that $|u(t_{M})-u_{M}|=O((\Delta t)^{k+1-\alpha})$ in the cases of $1\le i\le k\le3$.
\end{remark} | 154 | 54,748 | en |
train | 0.45.17 | \section{Numerical experiments}\label{numexpsection}
In this section, we utilize formula \eqref{Dki1} to approximate the following equations in Example \ref{ex:4} and Example \ref{ex:5}, and we prescribe the starting values exactly. Since in practical computation, the starting values are normally obtained by numerical computation in advance.
\begin{example}
\label{ex:4}
We consider the linear equation
\begin{equation}
\label{eq:4.1}
\begin{cases}
&{^C}D^{\alpha}u(t)=\lambda u(t)+f(t), \hspace{0.3cm} t\in(0,1], \\
&u(0)=u_{0}
\end{cases}
\end{equation}
with $0<\alpha<1$. The exact solution is $u(t)=e^{-t}\in C^{\infty}[0,1]$, and $
f(t)=-t^{1-\alpha}E_{1,2-\alpha}(-t)-\lambda e^{-t}\in C[0,1]\cap C^{\infty}(0,1]$, where the Mittag-Leffler function \cite{Podlubny:1999} is defined by
\begin{equation*}
E_{\alpha, \beta}(t)=\sum_{k=0}^{\infty}\frac{t^{k}}{\Gamma(\alpha k+\beta)}, \hspace{0.5cm} \alpha>0,\hspace{0.1cm}\beta>0.
\end{equation*}
\end{example}
\begin{figure}
\caption{The boundary of the stability region for different $\alpha$ and $\lambda$.}
\label{F1:subfig1}
\label{F1:subfig2}
\label{F1:subfig3}
\label{F1:subfig4}
\label{fig:F1}
\end{figure}
\begin{table}[ht!]
\centering
\caption{The error accuracy and convergence rate of $|u(t_{M})-u_{M}|$ in Example \ref{ex:4} for different $\alpha$ and $\lambda$.}
\label{ta:3}
\footnotesize
\begin{tabular*}{\textwidth}{@{\extracolsep{0.1cm}}c@{\extracolsep{0.4cm}}c@{\extracolsep{0.5cm}}l@{\extracolsep{0.5cm}}l@{\extracolsep{0.8cm}}l@{\extracolsep{0.8cm}} l@{\extracolsep{0.8cm}}l @{\extracolsep{0.8cm}}l@{\extracolsep{0.8cm}}l@{\extracolsep{0.1cm}} }
\toprule
$\alpha$&$\lambda$ & $M$ &\multicolumn{2}{l}{$(k, i)=(1,1)$} & \multicolumn{2}{l}{$(k,i)=(2, 1)$}&\multicolumn{2}{l}{$(k,i)=(2, 2)$} \\
\cline{4-9}
&&&$|u(t_{M})-u_{M}|$&rate&$|u(t_{M})-u_{M}|$ &rate &$|u(t_{M})-u_{M}|$ &rate \\
\midrule
0.5 & -1 & 128 & 1.59038E-04 & - & 1.60073E-07 & - & 1.34983E-07 & - \\
& & 256 & 5.53407E-05 & 1.52 & 2.86635E-08 & 2.48 & 2.37399E-08 & 2.51 \\
& & 512 & 1.93502E-05 & 1.52 & 5.11315E-09 & 2.49 & 4.18230E-09 & 2.50 \\
& & 1024 & 6.78837E-06 & 1.51 & 9.09687E-10 & 2.49 & 7.37621E-10 & 2.50 \\
& & 2048 & 2.38698E-06 & 1.51 & 1.61541E-10 & 2.49 & 1.30187E-10 & 2.50 \\
\midrule
0.3 & $20\times e^{\frac{i\pi\alpha}{2}}$ & 128 & 1.34901E-06 & - & 2.69029E-09 & - & 1.08970E-09 & - \\
& & 256 & 3.73401E-07 & 1.85 & 4.44355E-10 & 2.60 & 1.62447E-10 & 2.75 \\
& & 512 & 1.04588E-07 & 1.84 & 7.14993E-11 & 2.64 & 2.44242E-11 & 2.73 \\
& & 1024 & 2.96205E-08 & 1.82 & 1.13426E-11 & 2.66 & 3.69238E-12 & 2.73 \\
& & 2048 & 8.47625E-09 & 1.81 & 1.78666E-12 & 2.67 & 5.57796E-13 & 2.73 \\
\midrule
0.9 & $1000\times e^{\frac{i\pi\alpha}{2}}$ & 128 & 7.84215E-07 & - & 3.94997E-09 & - & 3.83098E-09 & - \\
& & 256 & 3.63985E-07 & 1.11 & 9.18845E-10 & 2.10 & 8.91101E-10 & 2.10 \\
& & 512 & 1.69345E-07 & 1.10 & 2.14033E-10 & 2.10 & 2.07571E-10 & 2.10 \\
& & 1024 & 7.88889E-08 & 1.10 & 4.98901E-11 & 2.10 & 4.83852E-11 & 2.10 \\
& & 2048 & 3.67748E-08 & 1.10 & 1.16334E-11 & 2.10 & 1.12827E-11 & 2.10 \\
\midrule
0.98 & $500 i$ & 128 & 2.55224E-06 & - & 1.32455E-08 & - & 1.31778E-08 & - \\
& & 256 & 1.25624E-06 & 1.02 & 3.25627E-09 & 2.02 & 3.23963E-09 & 2.02 \\
& & 512 & 6.18899E-07 & 1.02 & 8.01689E-10 & 2.02 & 7.97599E-10 & 2.02 \\
& & 1024 & 3.05047E-07 & 1.02 & 1.97518E-10 & 2.02 & 1.96512E-10 & 2.02 \\
& & 2048 & 1.50388E-07 & 1.02 & 4.86819E-11 & 2.02 & 4.84347E-11 & 2.02 \\
\bottomrule
\end{tabular*}
\end{table} | 2,200 | 54,748 | en |
train | 0.45.18 | \begin{table}[ht!]
\centering
\caption{The error accuracy and convergence rate of $|u(t_{M})-u_{M}|$ in Example \ref{ex:4} for different $\alpha$ and $\lambda$.}
\label{ta:4}
\footnotesize
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}c@{\extracolsep{0.4cm}}c@{\extracolsep{0.5cm}}l@{\extracolsep{0.5cm}}l@{\extracolsep{0.7cm}}c@{\extracolsep{0.7cm}}l@{\extracolsep{0.7cm}}c @{\extracolsep{0.7cm}}l@{\extracolsep{0.7cm}}c@{\extracolsep{0.1cm}} }
\toprule
$\alpha$&$\lambda$ & $M$ &\multicolumn{2}{l}{$(k, i)=(3,1)$} & \multicolumn{2}{l}{$(k,i)=(3, 2)$}&\multicolumn{2}{l}{$(k,i)=(3, 3)$} \\
\cline{4-9}
&&&$|u(t_{M})-u_{M}|$&rate&$|u(t_{M})-u_{M}|$ &rate &$|u(t_{M})-u_{M}|$ &rate \\
\midrule
0.5 & -1 & 128 & 1.04028E-09 & - & 9.41107E-10 & - & 9.99698E-10 & - \\
& & 256 & 9.23186E-11 & 3.49 & 8.25515E-11 & 3.51 & 8.86817E-11 & 3.49 \\
& & 512 & 8.18229E-12 & 3.50 & 7.25575E-12 & 3.51 & 7.85577E-12 & 3.50 \\
& & 1024 & 7.25420E-13 & 3.50 & 6.37490E-13 & 3.51 & 6.92002E-13 & 3.50 \\
& & 2048 & 6.82232E-14 & 3.41 & 5.34572E-14 & 3.58 & 5.85643E-14 & 3.56 \\
\midrule
0.3 & $20\times e^{\frac{i\pi\alpha}{2}}$& 128 & 1.73101E-11 & - & 1.03070E-11 & - & 1.53161E-11 & - \\
& & 256 & 1.33740E-12 & 3.69 & 7.55245E-13 & 3.77 & 1.18503E-12 & 3.69 \\
& & 512 & 1.03673E-13 & 3.69 & 5.58325E-14 & 3.76 & 9.13054E-14 & 3.70 \\
& & 1024 & 6.13266E-15 & 4.08 & 4.40825E-15 & 3.66 & 7.52355E-15 & 3.60 \\
& & 2048 & 1.20505E-15 & 2.35 & 6.86635E-16 & 2.68 & 1.12983E-15 & 2.74 \\
\midrule
0.9 & $1000\times e^{\frac{i\pi\alpha}{2}}$ & 128 & 2.28884E-11 & - & 2.24089E-11 & - & 2.26564E-11 & - \\
& & 256 & 2.65645E-12 & 3.11 & 2.60061E-12 & 3.11 & 2.62969E-12 & 3.11 \\
& & 512 & 3.09069E-13 & 3.10 & 3.02593E-13 & 3.10 & 3.05970E-13 & 3.10 \\
& & 1024 & 9.71032E-07 & -21.58 & 4.20376E-06 & -23.73 & 4.42542E-07 & -20.46 \\
& & 2048 & 6.28199E+34 & -135.57 & 1.41894E+35 & -134.63 & 1.09390E+34 & -134.18 \\
\midrule
0.98 & $500 i$ & 128 & 7.76488E-11 & - & 7.73754E-11 & - & 7.75095E-11 & - \\
& & 256 & 9.52734E-12 & 3.03 & 9.49401E-12 & 3.03 & 9.51047E-12 & 3.03 \\
& & 512 & 9.64788E-07 & -16.63 & 1.77544E-06 & -17.51 & 1.37083E-06 & -17.14 \\
& & 1024 & 3.88686E-14 & 24.57 & 1.83022E-13 & 23.21 & 1.31363E-13 & 23.31 \\
& & 2048 & 1.75624E-14 & 1.15 & 1.70830E-14 & 3.42 & 1.66951E-14 & 2.98 \\
\bottomrule
\end{tabular*}
\end{table}
Figures \ref{F1:subfig1}-\ref{F1:subfig4} plot the truncated boundary locus curves $\sum_{n=0}^{6000}\omega_{n}^{(k,i)}e^{in\theta}~(0\le \theta\le 2\pi)$ in the cases of $1\le i\le k\le 3$ for different $\alpha\in (0,1)$. It is known from Theorem \ref{the:stabilityr} that the stability regions of methods \eqref{DkiTest} lie outside the corresponding curves. We introduce the points $z_{n}:=\lambda (\Delta t_{n})^{\alpha}$ for $1\le n\le 5$, where $\Delta t_{n}=1/2^{n+6}$ correspond to different time stepsize. Table \ref{ta:3} and Table \ref{ta:4} list the global error $e_{M}=u(t_{M})-u_{M}$ in Example \ref{ex:4}, where $t_{M}=M\Delta t=1$ is fixed and $M=2^{j}$ for $7\le j\le 11$, $u(t_{M})$ and $u_{M}$ are the exact solution and the computed solution, respectively.
By the comparison of Figures \ref{F1:subfig1}-\ref{F1:subfig4} and Tables \ref{ta:3}-\ref{ta:4}, we can see the influence of the stability of the numerical methods on the error accuracy. In Figure \ref{F1:subfig1}, the points $z_{n}$ for $1\le n\le 5$ all lie in the stability regions in the case of $\alpha=0.5$ and $\lambda=-50$, in which situation the reliable accuracy is obtained, and it is observed that $|e_{M}|=O(\Delta t^{k+1-\alpha})$ in Table \ref{ta:3}-\ref{ta:4}. In Figure \ref{F1:subfig2}-\ref{F1:subfig3}, $z_{n}$ are choose on the half line with angle $\frac{\pi\alpha}{2}$ and different $\lambda$, it is observed that when all $\{z_{n}\}_{n=1}^{5}$ fall out of the instability region (cf. Figure \ref{F1:subfig2}), correspondingly, as shown in Table \ref{ta:3}-\ref{ta:4}, the the global error $e_{M}$ agrees with the expectation of the accuracy. On the other hand, due to points $z_{4}$ and $z_{5}$
outside the stability regions of $k=3$ (cf. Figure \ref{F1:subfig3}), the perturbation errors are magnified and accumulated significantly, which are shown in Table \ref{ta:3}-\ref{ta:4} as well. In Figure \ref{F1:subfig4}, $z_{n}$ are choose on the imaginary axis with imaginary number $\lambda$, according to Theorem \ref{th:Api}, all the $z_{n}$ belongs to the stability region of the method methods \eqref{DkiTest} in the case of $k=1, 2$, and the error accuracy and the convergence order are obtained (cf. Table \ref{ta:3}). As a counter example, when $z_{3}$ doesn't belong to the stability region for $\alpha=0.98$ in Figure \ref{F1:subfig4}, the corresponding error $e_{M}$ shown in Table \ref{ta:4} can't ensure the desirable accuracy. In fact, it can be observed that for $k=3$, methods \eqref{DkiTest} don't possess $A(\frac{\pi}{2})$-stability when $\alpha$ tends to $1$, which appears to be predictable, since it is well known that BDF3 method for ODEs is not $A(\frac{\pi}{2})$-stable.
\begin{example}
\label{ex:5}
Consider the nonlinear equation
\begin{equation}\label{eq:4.2}
\begin{cases}
&{^C}D^{\alpha}u(t)=-u^{2}+f(t), \hspace{0.3cm}t\in (0,1] \\
&u(0)=u_{0}
\end{cases}
\end{equation}
with exact solution $u(t)=e^{\mu t}$ and source function
$f(t)=\mu t^{1-\alpha}E_{1,2-\alpha}(\mu t)+e^{2\mu t}$.
\end{example}
\begin{figure}
\caption{Errors and convergence orders of $|e_{M}
\label{F4:subfig1}
\label{F4:subfig2}
\label{F4:subfig3}
\label{F4:subfig4}
\label{fig:F4}
\end{figure}
\begin{figure}
\caption{Errors and convergence orders of $|e_{M}
\label{F5:subfig1}
\label{F5:subfig2}
\label{F5:subfig3}
\label{F5:subfig4}
\label{fig:F5}
\end{figure}
Figure \ref{fig:F4} and \ref{fig:F5} plot the global error $|e_{M}|=|u(t_{M})-u_{M}|$ in Example \ref{ex:5} for different $\mu$ and $\alpha$, where $t_{M}=1$ is fixed and $\Delta t=1/M$ with $M=2^{j},~2\le j\le 11$. It is observed that $|e_{M}|=O(\Delta t^{k+1-\alpha})$ in the cases of $1\le i\le k\le 3$, when using discretisation formula \eqref{eq:nnonliode} in combination with Newton's method for the nonlinear equation behind the implicit method. | 3,159 | 54,748 | en |
train | 0.45.19 | \section{Conclusions}\label{concl}
We have proposed a new higher-order approximation method for solving time-fractional initial value models of order $0<\alpha<1$. Furthermore, the local truncation error in terms of solution possessing sufficient smoothness is derived and demonstrated. Additionally, the stability and convergence analyses of the proposed method are provided in details, which also motivate the relevant research on applications to time-fractional partial differential equations.
\end{document} | 106 | 54,748 | en |
train | 0.46.0 | \begin{document}
\title{Monitoring Hyperproperties with Circuits hanks{The authors were supported by the projects `Open Problems in the Equational Logic of Processes’ (OPEL) (grant No 196050-051) and `Mode(l)s of Verification and Monitorability' (MoVeMent) (grant No~217987) of the Icelandic Research Fund, and `Runtime and Equational Verification of Concurrent Programs' (ReVoCoP) (grant No 222021), of the Reykjavik University Research Fund. Luca Aceto's work was also partially supported by the Italian MIUR PRIN 2017 project FTXR7S IT MATTERS `Methods and Tools for Trustworthy Smart Systems'.}
\begin{abstract}
This paper presents an extension of the safety fragment of Hennessy-Milner Logic with recursion over sets of traces, in the spirit of Hyper-LTL. It then introduces a novel monitoring setup that employs circuit-like structures to combine verdicts from regular monitors.
The main contribution of this study is the definition of the monitors and their semantics, as well as a monitor-synthesis procedure from formulae in the logic that yields ‘circuit-like monitors’ that are sound and violation complete over a finite set of infinite traces.
\end{abstract}
\section{Introduction}
The field of runtime verification concerns itself with providing methods for checking whether a system satisfies its intended specification at runtime. This runtime analysis is done through a computing device called a \textit{monitor} that observes the current run of a system in the form of a trace \cite{Bartocci2018,francalanza_found_Run_Moni}. Runtime verification has recently been extended to the setting of concurrent systems
\cite{AcetoPOPL19,Bocchi_Tuosto_DistInteraction,Cassar2017ReliabilityAF,choreogr_monitors_as_mem}
with several attempts to specify properties over sets of traces, and to introduce novel monitoring setups \cite{Agrawal2016RuntimeVO,complex_monitor_hyper,monitor_hyper}. A centerpiece in this line of work has been the specification logic Hyper-\ltl \cite{hyperproperties}. Intuitively Hyper-\ltl allows for existential and universal quantification over a set of traces (which describes the set of observed system runs). The properties over one trace are stated in \ltl, with free trace variables, and then made dependent on properties of other traces via the quantification that binds the trace variables.
We define the linear-time specification logic \hyperhml, as a counterpart to Hyper-\ltl, building on
previous studies of monitorability and monitor synthesis for \muhml \cite{AcetoPOPL19,hml_monitors}, which are necessary for the kind of correctness and complexity guarantees we aim to achieve in this work.
However, just like Hyper-\ltl, \hyperhml can define dependencies over different traces, which intuitively causes extra delays in the processing of traces as the properties observed on one of them can impact what is expected for another. For example, if a property requires that an event of a trace is compared against an event occurring in all other traces then the processing cost of this event becomes dependent on the number of traces.
In this approach, we keep the processing-at-runtime cost (as defined in \cite{rabin_real_time_comp}) minimal by restricting the type of properties verified to a natural fragment of \hyperhml, but applying no assumptions on the system under scrutiny.
This comes in contrast with the existing research, where the runtime verification of such properties is dealt with via a plethora of modifications and assumptions made over the monitoring setup, such as being able to restart an execution or having access to all executions of a system.
Our monitor setup is engineered for the studied fragment of the specification language, by utilizing circuit-like structures to combine verdicts over different traces. The fragment of the logic restricts the amount of quantification that can be applied to the properties of individual traces and thus limits the dependencies between them. This naturally induces circuits with monitors from \cite{AcetoPOPL19} as input nodes and simple kinds of gates at the higher levels, with the resulting structure having constant depth with respect to the corresponding formula, which is considered efficient in the field of parallel computation \cite{complexity_small_depth_circuits}. Thus, each step taken by such a monitor in response to an event of the system under scrutiny takes constant time, which makes the monitors `real time' in the sense of \cite{rabin_real_time_comp}.
\section{The logic}
Our logic is defined in the style of Hyper-\ltl as presented in \cite{hyperproperties}. The quantification among traces remains the same, but the language in which local trace properties are stated is \muhml.
We consider the following restriction to a multi-trace \shml logic (the \textit{safety} fragment of \muhml \cite{AcetoPOPL19}), with no alternating quantifiers, called \ellihml. We can similarly define the \chml (co-safety) fragment, and the \hml fragment.
\begin{definition}
Formulae in $\textsc{Hyper}^1\text{-}\textsc{sHML}\xspace$ are constructed by the following grammar:
\begin{align*}
\varphi \in \textsc{Hyper}^1\text{-}\textsc{sHML}\xspace& ::= \exists_{\pi} \psi &\mid&~ \forall_{\pi} \psi &\mid&~ \varphi \sqcup \varphi &\mid&~ \varphi \sqcap \varphi
\end{align*}
where
$\psi$ stands for a formula in \shml and $\pi$ is a trace variable from an infinite suppy of trace variables $\mathcal{V}$.
$\sqcup \text{ and }\sqcap$ stand for the regular $\vee$ and $\wedge$ boolean connectives, only usable at the top syntax level.
Although the syntactic distinction is cosmetic, it allows us to keep the synthesis function in Definition \ref{def:synthesis} clearer.
\end{definition}
\textit{Semantics} The semantics of \hyperhml is given over a finite set of infinite traces $T$ over \act and it is a natural extension of the linear-time semantics of \muhml. The existential and universal quantification happens via the trace variable $\pi$ which ranges over the traces in $T$. The extension of the \muhml linear-time semantics from \cite{AcetoPOPL19} to the \hyperhml semantics is done in the style of Hyper-\ltl.
This semantics applies to \ellihml, which is a fragment of \hyperhml.
We only consider \textit{closed} formulae in \ellihml and for these we use the standard notation $T \models \varphi$ to mean that a set of traces $T$ satisfies $\varphi$ (and similarly for $ T \not \models \varphi$).
\begin{example}\label{ex:1} The \ellihml formula $ \forall_{\pi} [a]\mathtt{ff} \sqcap \exists_{\pi} [b] ( max~x. ([a]\mathtt{ff} ~\wedge ~[b] x))$, over the set of actions $\{a,b\}$, states that for any set of traces $T$, none of the traces in $T$ start with $a$, and $b^\omega \in T$.
\end{example} | 1,794 | 7,248 | en |
train | 0.46.1 | \section{The monitors}
The intuition behind our monitor design is the following (we recommend following this intuition along with the example given in Figure \ref{fig:circuit_monitor}).
Over a finite set of traces $T$ we instrument a circuit-like structure.
Each trace $t \in T$ is assigned a fixed set of regular monitors that correspond to the properties in \shml to be verified.
These regular monitors are connected with simple gates which evaluate to $yes$, $no$ or $end$ based on the verdicts produced by their associated regular monitors. Once some of these gates start evaluating to verdicts, they communicate with more complex gates, connected in a circuit-like graph, which propagate input verdicts though logic operations until the root node of the circuit reaches a verdict as well. The formal definition of a circuit monitor is given in the style of computational complexity circuits \cite[Definition~1.10]{circuit_complexity}.
\begin{definition}
The language $\text{C}\textsc{mon}_k$ of $k$-ary monitors, for $k >0$, is given through the following grammar:
\begin{align*}
&M \in \text{C}\textsc{mon}_k
::= \bigvee [m]_k &&|~~ \bigwedge [m]_k &&| ~~ M \vee M &&|~~ M \wedge M
\\
& m ::=~~~ yes ~| ~no ~|~end &&|~~ a.m,~ a\in \act &&|~~ m +n &&|~~ rec ~x.m &&|~~ x
\end{align*}
\text{C}\textsc{mon}\xspace is the collection of infinite \textit{sequences} $(M_i)_{i \in \nat}$ of terms that are generated by substituting $k = i, \forall i \in \nat$, in a term $M
$ in $\text{C}\textsc{mon}_k$.
\end{definition}
We use $M,M' \ldots$ to denote the monitors (infinite sequences of terms generated by the first line of this grammar), and refer to them as circuit monitors, and $m_1, m_2 \ldots$ to denote the regular monitors described by the second line.
The notation $[m]_k$ corresponds to the parallel dispatch of $k$ identical regular monitors $m$, where $k = |T|$, with $T =\{t_1, \ldots ,t_k \}$.
Given a monitor $M \in \text{C}\textsc{mon}\xspace$, for some $k >1$, we will call each syntactic sub-monitor of $M$ a gate. For example, we have inductively that over the monitor $M' \vee M''$ we have the gates $M' \vee M''$ and all gates contained in monitors $M'$, and $M''$, while for the monitor $\bigvee [m]_k$ we have the gates $\bigvee [m]_k$ and gates $m_{[i]}$ for $i \in \{ 1, \ldots, k \}$.
For $M \in \text{C}\textsc{mon}\xspace$ we define a set of \textit{program variables} $G_M$, where one variable $g_{M'}$ is assigned to each gate $M'$ of $M$.
For readability purposes we will be omitting the naming $g$ of the program variables and call them by the name of the gate they represent.
We use
$m_{[i]}$ to mean the regular monitor $m$ instrumented over the trace $t_i$.
It is important here to see that $g_{m_{[i]}}$ will be the \textit{name} of the gate assigned to one such monitor and stays unchanged while the actual monitor advances its computation as trace events are read.
This will be clarified later, through the instrumentation rules.
A program variable related to gate $M$, can be assigned the following values: $yes$, $no$, $end$, and $j$, with $j \in \{ 0, \dots, 2^{(\ell+1)}-1 \}$, $\ell$ being the number of immediate syntactical sub-monitors of gate $M$.
Number $j$ is encoded in binary, and is used to carry the information of which sub-gates have given some verdict (this means that the encoding of $j$ has $\ell +1$ bits).
The value of the $\ell +1$-th bit of $j$ is reserved to encode that one of the sub gates has outputted an $end$. The information that $j$ carries is very important for the evaluation of a gate, as often this evaluation depends on the verdicts of more that one sub-gate, as well as what these verdicts are (see Figure \ref{fig:circuit_monitor}). A variable $g_m$ can only take the values $yes$, $no$ and $end$, produced by the relevant monitor instrumented over a trace.
A \textbf{\textit{configuration}} of monitor $M \in \text{C}\textsc{mon}\xspace$, for some $k>1$, is an array $s_M$ containing a value for all program variables $g$ of $M$. We denote the set of all configurations for a monitor $M$ as $\mathcal{S}_M$. We use the notation $s[M\backslash i]$ to denote the update of a configuration $s$ where gate $M$ stores some value $j$ to one where the $i$-th coordinate of $j$ is $0$, while all other variables have the value they had in configuration $s$. Similarly, we use the notation $s[M\backslash end_i]$ to refer to a configuration where the update $s[M\backslash i]$ has taken place \textit{and} the value of the $\ell +1$-th bit of $j$ is set to $1$. We also use the notation $s[\sfrac{v}{M}]$ with $v \in \{yes,no,end\}$, to mean a configuration where the value of the variable for gate $M$ is updated to $v$.
All gate variables in a circuit monitor are initialized to $2^{\ell}-1$ (a sequence of $\ell$-many zeros), to represent that all sub-gates are waiting to give some output and $s_{M_{init}}$ stands for the initial configuration of $M$. Since $M$ is a family of circuits, we have that the initial configuration of each monitor $M_i$ in the family corresponds to a different initial configuration $s_{M_{i-init}}$.
\begin{example}
Figure \ref{fig:circuit_monitor}, is an example of a circuit monitor and its evaluation.
\end{example}
\begin{figure}
\caption{The circuit monitor for the formula from Example \ref{ex:1}
\label{fig:circuit_monitor}
\end{figure}
\begin{figure}
\caption{\label{tab:sos_rules}
\label{tab:sos_rules}
\end{figure}
\textit{Semantics}
The semantics of a regular monitors is as presented in \cite{AcetoPOPL19}. Each regular monitor corresponds to an LTS, and a transition labeled with $a \in \act$ corresponds to a regular monitor observing the event $a$ when instrumented with a system $p$ that produces it.
The semantics of a circuit monitor is given as a transition relation $\xrightarrow[\text{}]{} \subseteq$ $\mathcal{S}_M \times \mathcal{S}_M$ and the instrumentation $\triangleleft$ takes place over a set of regular monitors $\overrightarrow{m}$ instrumented over a set of traces $T$, denoted $M(T)$.
We define $M(T): = s_{M_{|T|-init}} \triangleleft \overrightarrow{m}_{[i]} \triangleleft T$, where $\overrightarrow{m}$ is the set of regular monitors that occur in $M$, and $\overrightarrow{m}_{[i]}$ is $ \overrightarrow{m}$, instrumented over the trace $t_i \in T.$ When $m$ is a regular monitor then $\triangleleft$ stands for the existing instrumentation relation from \cite{AcetoPOPL19}.
The transition and instrumentation relations are defined as the least ones that satisfy the axioms and rules in Figure \ref{tab:sos_rules}. Due to lack of space, we only include the rules giving the semantics of the $\bigvee [m]_k$ monitor. Those for the other operators follow the same structure. The proof in Appendix \ref{appendix:proof_viol_comp} could help with the understanding of the more intricate instrumentation rules.
A monitor is required to be \textit{correct} with respect to some specification formula $\varphi$. The notions of correctness we use in this work are defined below.
\begin{definition}
Given a monitor $M \in$ \text{C}\textsc{mon}\xspace , and a set of traces $T$.
\begin{itemize}
\item $M$ \textbf{rejects} $T$ (resp. \textbf{accepts} $T$) denoted $rej(M,T)$ (resp. $acc(M,T)$) iff $M(T) \rightarrow^* s \triangleleft \overrightarrow{n}\triangleleft T'$ for some $s,\overrightarrow{n},~ T'$, where $s[M]=no$ (resp. $s[M]=yes$).
\item Given a formula $\varphi \in$ \hyperhml, $M$ is \textbf{sound} for $\varphi$ if $\forall T$, $acc(M,T) \implies T \models \varphi$, and $rej(M,T) \implies T \not \models \varphi$.
\item $M$ is \textbf{violation complete} for $\varphi$ if $\forall T$, $T\not \models \varphi \implies rej(M,T)$.
\end{itemize}
\end{definition}
\textit{Synthesis:}
Given a formula $\varphi$ in \ellihml, We synthesize a circuit monitor $M$ through the following recursive function $Syn(-) :$ \ellihml $\rightarrow$ \text{C}\textsc{mon}\xspace.
\begin{definition}[Circuit Monitor Synthesis]\label{def:synthesis}
\begin{align*}
&Syn(\exists_{\pi} \varphi)= \bigvee [m(\varphi)]_{k} &&~~ Syn(\forall_{\pi} \varphi)= \bigwedge [m(\varphi)]_{k} \\
&Syn( \varphi_1 \sqcup \varphi_2)= Syn(\varphi_1) \vee Syn(\varphi_2)
&&~~ Syn( \varphi_1 \sqcap \varphi_2)= Syn(\varphi_1) \wedge Syn(\varphi_2)
\end{align*}
Where $m(-)$ is the monitor synthesis function for \shml defined in \cite{AcetoPOPL19}.
\end{definition}
\begin{proposition}\label{{prop:viol_comp}} Given a formula $\varphi$ in \ellihml, we have $Syn(\varphi)$ is a sound and violation-complete monitor for $\varphi$.
\end{proposition}
\begin{proof}
The proof is by induction on the structure of $\varphi$. We present here a characteristic case and give more details for some of them in the Appendix \ref{appendix:proof_viol_comp}.
Assume that $ \varphi = \exists_{\pi} \psi$, with $ \psi \in \shml$ and that we have a set of traces $T$ s.t. $T \not \models \varphi$.
From the semantics of \ellihml, we have that $t_i \not\models \psi$, for all traces $t_i$ in $T$.
However $\psi \in \shml$ and thus from \cite{AcetoPOPL19} we get that $m_{\psi}$ is a violation complete monitor for $\psi$.
This means that for all $t_i \in T$, there exist $t_i' \in \act^*$ and $t_i'' \in \act^{\omega}$, such that $t_i = t_i'.t_i''$, such that the monitor $m_{\psi}$ rejects $t_i'$.
From the rules in Figure \ref{tab:sos_rules} we see that each gate $g_{m_{\psi[i]}}$ will reach the value $no$ as enough events over the trace $t_i'$ will occur. I.e. $s_M \triangleleft \overrightarrow{m_{\psi_{[i]}}} \triangleleft T \rightarrow^* s_M \triangleleft \overrightarrow{m}_{[i]}[\sfrac{no}{m_{[i]}}] \triangleleft T[\sfrac{t_i''}{t_i}]$, witch propagates to the evaluation of $g_{m_{[i]}}$ to $no$, for all $i$. We now study the transitions $s_M[\sfrac{no}{g_{m_{\psi}[i]}}]$ since those can be then composed with this instrumentation via the fourth instrumentation rule. Applying the SOS rules yields that the update $\backslash i$ takes place for all $i$ at the gate $\bigvee [m]_k$ which means that the value of $j$ stored in it becomes $0$. This finally yields that the value of the final gate $\bigvee [m]_k$ becomes $0$, i.e. $s_M [\sfrac{no}{g_{m_{[i]}}}~ \forall i] \rightarrow s_M [\sfrac{no}{\bigvee [m]_k}]$. Since this transition can be composed with the discussed instrumentation we have that $s_M \triangleleft \overrightarrow{m_{\psi_{[i]}}} \triangleleft T \rightarrow s_M[\sfrac{no}{g_{\bigvee [m_{\psi}]_{[i]}}}]\triangleleft \overrightarrow{n} \triangleleft T'$ for some $\overrightarrow{n}$ and $T$ and we are done. \qed
\end{proof}
\subsection{Runtime costs}
The monitor synthesis in Definition \ref{def:synthesis} provides a family of circuits that can be instrumented appropriately on an arbitrary set of traces to analyze the events occurring in them. Ideally, the runtime cost of monitoring resulting from our constructions should be bounded by a constant that does not depend on the parameters of the system (such as the number of available traces, or of the events observed so far) \cite{rabin_real_time_comp}. In this way, if a monitor is launched along with the system components, it will only induce a feasible computational overhead.
We already know that the regular monitors instrumented with individual traces analyze the system events they observe with a constant overhead \cite{hml_monitors}. Regarding the computational cost of the circuit part, since we are given $k$ many traces, it must be that the necessary computation performed from a circuit monitor can be performed in parallel, distributed over the components that produced the traces in the first place. This means that we can only concern ourselves with the \textit{circuit complexity} \cite{circuit_complexity} of a given monitor, which encapsulates the parallel processing power necessary for its evaluation.
We now observe the synthesis function. There, a formula $\varphi$ in \ellihml will be turned into a family of circuit monitors where, for each connective of the original formula $\varphi$, the output monitor increases in size based on the size for the monitors of the sub-formulae of $\varphi$. However, for each connective of the formula, the \textit{depth} of the circuit is only increased by $1$ which means that the output circuit monitor has a depth bounded by the size of the formula $\varphi$. Since the gates of the output monitor can have either a fixed amount of sub-gates ($\vee, \wedge$), or $k$ many ($\bigvee, \bigwedge$), we have that the output circuit is in the complexity class AC$^0$ \cite{circuit_complexity}. Thus, the monitor only adds a constant computational overhead when executed over the computational resources of the distributed components of the system. | 3,810 | 7,248 | en |
train | 0.46.2 | \section{Conclusion and future work}
We expect that the fragment \ellihml is maximal with respect to violation completeness, which means that any monitor in \text{C}\textsc{mon}\xspace is monitoring for a formula in \ellihml.
However, the ultimate goal of this work is to extend the collection of monitorable properties by allowing alternating quantifiers in the syntax.
This is a very important aspect of any work in this field, as the more interesting hyperproperties, such as the property ``at all times, if one trace encounters the event $p$ then all traces do so as well'' which is a necessary component for the expression of properties such as noninference \cite{temp_logic_hyperprop,possibilistic_prop}, require alternation of quantifiers.
A way to tackle this would be to project such properties into the \ellihml fragment. However this procedure is not formally yet defined, or trivial and one could argue that since every hyperproperty has been shown (\cite{hyperproperties}) to be the intersection of a liveness and a safety hyperproperty, (and since liveness and safety properties are widely accepted as independent \cite{alpernBowenSchneiderLiveandSafe}), an elimination of alternating quantifiers can only take place in very few cases.
Thus, our main purpose is to extend the logic and the consequent monitors in order to express and monitor for the most general class of such properties.
The main objective of the logical fragment we give here is to establish a formal baseline which we will attempt to extend in future work.
Our approach to an extension would be to allow a notion of synchronization rounds among the regular monitors (or equivalently a round of communication). This would enable more complex dependencies between traces, as now the properties required of a given trace can be impacted by the state of the ones monitored for on a different one. However, the analysis of communications among the monitors is a complicated extension, as their exact content plays a significant role to our insight over the system, as well as the processing at runtime cost. We plan to implement this therefore by utilizing dynamic epistemic logic \cite{dyn_epist_logic} in order to perform this extension formally and soundly.
\appendix
\section{Appendix: cases for the proof of violation completeness}\label{appendix:proof_viol_comp}
Here we give some more insight on the remaining cases of the violation completeness proof. First we highlight that the second base case of our proof, for formulae of the form $\forall_{\pi} \psi$ is completely analogous to the one we give and thus omitted.
We will here give an important lemma necessary for analyzing both remaining cases, and then present the high level details for the case of $\sqcap$. The intuition of the importance of the lemma is that the monitors $Syn(\varphi_1)$ and $Syn(\varphi_2)$ should not have their computation affected from the fact that they are run in parallel over a set of traces $T$.
\begin{lemma}
If
\begin{itemize}
\item $s_{M_1} \triangleleft \overrightarrow{m_1}[i] \triangleleft T\rightarrow s_{M_1}'\triangleleft \overrightarrow{m_1}[i]' \triangleleft T'$, and
\item $s_{M_2} \triangleleft \overrightarrow{m_2}[i] \triangleleft T\rightarrow s_{M_2}'\triangleleft \overrightarrow{m_2}[i]' \triangleleft T'$
\end{itemize}
then
\begin{itemize}
\item $s_{M_1 \vee M_2} \triangleleft \overrightarrow{m_{12}}[i] \triangleleft T\rightarrow s_{M_1 \wedge M_2}' \triangleleft \overrightarrow{m_{12}}[i]' \triangleleft T'$, and
\item $s_{M_1 \wedge M_2} \triangleleft \overrightarrow{m_{12}}[i] \triangleleft T\rightarrow s_{M_1 \wedge M_2}' \triangleleft \overrightarrow{m_{12}}[i]' \triangleleft T'$,
\end{itemize}
where $\overrightarrow{m_{12}} = \overrightarrow{m_{2}} \cup \overrightarrow{m_{2}}$ and $\overrightarrow{m_{12}}' = \overrightarrow{m_{2}}' \cup \overrightarrow{m_{2}}'$ respectively.
\end{lemma}
\begin{proof}
We note here that a configuration for $s_{M_1 \vee M_2}$ is identical to one for $s_{M_1 \wedge M_2}$ except the root variable, as all other variables they both contain are $s_{M_1}' \cup s_{M_2}'$.
The key aspect of this proof is the third rule of the instrumentation relation. There we can see that in order for a configuration instrumented over a set of regular monitors, instrumented over a set of traces, can only advance its computation, if all monitors instrumented over the same trace progress with their computation synchronously by reading the next trace event.
Thus, form the assumptions of this lemma we get that for all $j = \{1,\ldots r\}$, where $r$ is the total amount of different regular monitors occurring in $M_1$ and $M_2$ the premise of our rule is satisfied and thus the cumulative configuration of variables amounting for the union of variables of the two circuit monitors $M_1$ and $M_2$ (including the root variable), can perform the necessary transition to the new state, where all regular monitors (those both from $M_1$ and $M_2$) assigned to trace $t_i$ have processed the event $a$, and we are done. \qed
\end{proof}
Having the above lemma streamlines our inductive step for the rest of the cases.
Assuming a non-base-case formula in \ellihml we can clearly see that it must be of the form $\varphi = \varphi_1 \sqcap \varphi_2$ or $\varphi = \varphi_1 \sqcap \varphi_2$. We only analyze one of the two cases as they are symmetrical.
For any set of traces $T$, such that $T \not\models \varphi$, from the semantics of \ellihml, we have that $T \not\models \varphi_1$ and $T \not\models \varphi_2$. Since the synthesized monitor for $\varphi_1 \sqcap \varphi_2$ can reach a configuration where the values of the gates for $Syn(\varphi_1)$ and $Syn(\varphi_2)$ are the same as they would be for the individual monitors instrumented over $T$, and by inductive hypothesis (which guarantees that $Syn(\varphi_1)$ and $Syn(\varphi_2)$ are violation-complete) we have necessary conclusion by combining the two negative verdicts of the individual monitors via the semantics. \qed
\end{document} | 1,644 | 7,248 | en |
train | 0.47.0 | \begin{document}
\title[$\text{\sc Duality of Bochner spaces}$]
{Duality of Bochner spaces}
\author[S. Hiltunen]{Seppo\ I\. Hiltunen}
\address{Aalto University \vskip0mm$\hspace{2mm}$
Department of Mathematics and Systems Analysis \vskip0mm$\hspace{2mm}$
P.O.\ Box 11100 \vskip0mm$\hspace{2mm}$
FI-00076 Aalto \vskip0mm
Finland}
\email{seppo.i.hiltunen\,@\,aalto.fi}
\subjclass[2010]{Primary 46E40\kern0.37mm, 46A20\kern0.37mm, 46G10\kern0.37mm, 46E30\,; Secondary
28A20\kern0.37mm, 28B05\kern0.37mm, 46A16\kern0.37mm, 28C20}
\keywords{Bochner space, Banach space, duality, positive measure, positive
Radonian, vector measure, Lebesgue space, topological vector space, suitable
space, measurability, bounded variation, Dunford\,--\,Pettis.}
\begin{abstract} \renewcommand\lower1.05mm\hbox{$^+$}\infty{\lower.82mm\hbox{$^+$}\infty}\def\AbsLrs#1#2^#3{\raise.95mm\hbox{\font\≈=cmtt5\≈m\font\≈=cmssq5\≈v}\kern-.3mm\lower.25mm\hbox{\font\Â=cmitt10\ÂL}\kern0.25mm\kern0.#2mm\lower.#1mm\hbox{\raise.4mm\hbox{$^{#3}$}}\kern0.25mm}
We construct the generalized Lebesgue\,--\,Bochner spaces \math{
\AbsLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } for positive measures \math{\mu} and for
suitable real or complex topological vector spaces \math{\vPi} so that for \linebreak
\ú$
1<p<\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and Banachable \math{\vPi} with separable topology the
strong dual of the classical Bochner space \math{
\AbsLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } becomes canonically represented by \mathss37{
\AbsLrs23^p
\LHB{.2}{\KN{.2}^{^*}}\kern-0.3mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi^{\kern0.37mm\prime}_\sigma\kern0.07mm) }. Hence we
need no separability assumption of the norm topology of the strong dual \math{
\vPi^{\kern0.37mm\prime}_
{
\hbox{\font\≈=cmssi5\≈\char'031}}} of \mathss30{\vPi}. For \linebreak
\ú$
p=1\kern0.37mm$ and for suitably restricted positive measures \math{\mu} we even get
a similar result without any separability of the norm topology of the target
space \mathss30{\vPi}. For positive Radon measures on locally compact
topological spaces these results are essentially contained on pages
588\,--\,606 in R.\ E.\ Edwards' classical {\sl Functional Analysis\kern0.15mm}.
\end{abstract}
\maketitle
\Fsubhead Introduction and some preliminaries
Our main objective in this article is the following
\begin{Atheorem}\label{main Th}
Let \ú$\, 1 \le p < \lower1.05mm\hbox{$^+$}\infty ${\,\rm, }and let \ú$\,q =
(\kern0.37mm 1 - p\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm\big){}^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1} $ if \ú$\, p \not= 1
${\kern0.37mm\rm, }and \ú$\,q=\lower1.05mm\hbox{$^+$}\infty$ if \ú$\,p=1\kern0.37mm$.
Further{\kern0.15mm\rm, }
let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega${\,\rm, }and with \ú$\,\bosy K
\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \linebreak
\ú$\vPi\in\roman{BaS}\kern0.4mmps0(K)$ be such that {\,\rm(1)} or {\,\rm(2)} or
{\,\rm(3)} or {\,\rm(4)} or {\,\rm(5)} or {\,\rm(6)} below holds when
{\,\rm(\kern0.15mm\erm D\kern0.15mm)} means that $\,\mu$ is almost decomposable.
Also
let \ú$\, F = \mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) ${\KP1\rm, }and \ú$\, F\aar 1
= {\kern-0.63mm} $ \ú$\mvLrs03^q(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) $ if {\,\rm(1)} or
{\,\rm(2)} or {\,\rm(5)} or {\,\rm(6)} below holds,
otherwise letting \ú$\,
F\aar 1 = {\kern-0.63mm} $ \linebreak
$ \mvsLrs03^q(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)
\KP1 $. \ For \KP1 \vskip.5mm\centerline{$
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \seq{ \KP{1.2}
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.2mm\times\mathbb C\capss31\{\,(\kern0.37mm\smb X\kern0.15mm,\kern0.07mm t\kern0.37mm) :
\aall{x\in\smb X\kern0.15mm,\kern0.15mm y\in\smb Y}\,
t = \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KPt9\} :
\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\, } $} \inskipline{.5}0
then $\,\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm \in \Lis(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\dlbetss10\kern0.15mm)$ holds. In addition $\,
F\aar 1=\mvLrs03^q(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$ if {\,\rm(1)} or
{\,\rm(5)} holds. {\rm \inskipline14
(1)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and $\,\vPi$ is reflexive{\kern0.37mm\rm, \inskipline{.5}4
(2)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable
topology{\kern0.37mm\rm, \inskipline{.5}4
(3)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\mLrs42^1(\kern0.37mm\mu\kern0.15mm)$ is
a separable topology{\kern0.37mm\rm, \inskipline{.5}4
(4)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and a choice function $\,c\in
\Cal L\,(\kern0.37mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.15mm)\,,\kern0.15mm
\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.15mm{}^{}\Cal Omega\kern0.15mm)) $ exists{\kern0.37mm\rm, \inskipline{.5}4
(5)} \ $p\not=1$ and $\,\vPi$ is reflexive{\kern0.37mm\rm, \inskipline{.5}4
(6)} \ $p\not=1$ and $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable topology.
\end{Atheorem}
The proof is given on pages \pageref{Sec D}\,--\,\pageref{endmpf} below. Here
we first explain the notation appearing above, mentioning that we generally
utilize the notational convention explained in \cite[pp.\ 4\,--\,8]{HiDim}\,,
\cite[pp.\ 4\,--\,9]{SeBGN} and \cite[p.\ 1]{FKBGN}\,, and further to be
\q{polished} in \cite{Hif}\,. | 2,950 | 362,273 | en |
train | 0.47.1 | \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \seq{ \KP{1.2}
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.2mm\times\mathbb C\capss31\{\,(\kern0.37mm\smb X\kern0.15mm,\kern0.07mm t\kern0.37mm) :
\aall{x\in\smb X\kern0.15mm,\kern0.15mm y\in\smb Y}\,
t = \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KPt9\} :
\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\, } $} \inskipline{.5}0
then $\,\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm \in \Lis(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\dlbetss10\kern0.15mm)$ holds. In addition $\,
F\aar 1=\mvLrs03^q(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$ if {\,\rm(1)} or
{\,\rm(5)} holds. {\rm \inskipline14
(1)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and $\,\vPi$ is reflexive{\kern0.37mm\rm, \inskipline{.5}4
(2)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable
topology{\kern0.37mm\rm, \inskipline{.5}4
(3)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\mLrs42^1(\kern0.37mm\mu\kern0.15mm)$ is
a separable topology{\kern0.37mm\rm, \inskipline{.5}4
(4)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and a choice function $\,c\in
\Cal L\,(\kern0.37mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.15mm)\,,\kern0.15mm
\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.15mm{}^{}\Cal Omega\kern0.15mm)) $ exists{\kern0.37mm\rm, \inskipline{.5}4
(5)} \ $p\not=1$ and $\,\vPi$ is reflexive{\kern0.37mm\rm, \inskipline{.5}4
(6)} \ $p\not=1$ and $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable topology.
\end{Atheorem}
The proof is given on pages \pageref{Sec D}\,--\,\pageref{endmpf} below. Here
we first explain the notation appearing above, mentioning that we generally
utilize the notational convention explained in \cite[pp.\ 4\,--\,8]{HiDim}\,,
\cite[pp.\ 4\,--\,9]{SeBGN} and \cite[p.\ 1]{FKBGN}\,, and further to be
\q{polished} in \cite{Hif}\,.
Having \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} means that \math{\bosy K} is either the
topological field of real numbers or that of the complex ones. The underlying
sets of these fields are \math{\mathbb R} \nolinebreak and \nolinebreak \mathss38{
\mathbb C}, \linebreak
respectively. Then \math{\vPi\in\roman{BaS}\kern0.4mmps0(K)} means that \math{\vPi}
is a \erm Banach{\sl able\kern0.15mm}, i.e.\ a complete norm{\sl able\kern0.15mm} real or
complex topological vector space. Thus there is a compatible norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm}
on the underlying vector space \math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} such that \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\kern0.15mm,\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm) } is a norm{\sl ed\kern0.37mm} Banach space. Being
{\it compatible\kern0.37mm} here means that \math{
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm[\KP{1.1} 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.37mm\big] :
n\in\rbb Z^+\kern0.37mm\big\} } is a filter base for the filter $\neiBoo\vPi$ of
zero neighbourhoods. Above \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm[\KP{1.1} 0\,,\kern0.07mm
n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.37mm\big] } is the image of the closed interval \math{
[\KP{1.1} 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.37mm\big] } under the relational inverse \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm\inve} of \mathss31{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm}. Here \linebreak
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm$ is a function \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi
\to\lbb R_+} where \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} is the underlying set of vectors of \mathss31{
\vPi}. For \linebreak
$\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.37mm$ one may sometimes write \math{\|\,\xi\,\| }
for the value \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss11\xi} of \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} at \mathss34{\xi}.
For \mathss37{\vPi\aar 1\in\tvsps0(K)}, \,i.e.\ having \math{\vPi\aar 1} a
real or complex topological vector space with possibly non\kern0.37mm-\kern0.15mm Hausdorff
topology \mathss34{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\aar 1}, \,the exact construction of the space
$E=
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\aar 1)$ is given in
Constructions \ref{defi $L^p$}\,(\ref{simpL^p}) on page \pageref{simpL^p}
below. Here we informally explain the basic ideas
under the additional assumption that
$\vPi\aar 1$ is {\sl suitable\kern0.15mm} in the sense of
Definitions \ref{df suit} on page \pageref{df suit} below. Then
it suffices to consider one fixed
{\sl dominating\kern0.15mm} norm $\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ for $\vPi\aar 1\kern0.37mm$.
We consider functions
$x:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi$ such that on every set $A$ of finite measure, i.e.\
for $A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ it holds that outside some set $N$ of
measure zero, i.e.\ with $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ we have $x$
a pointwise limit of a sequence of simple functions, with convergence in the
sense of the topology $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$. In the case $p<\lower1.05mm\hbox{$^+$}\infty$ we then take
the subset of those $x$ such that the generally nonmeasurable
function \vskip.3mm\centerline{$
\Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x:{}^{}\Cal Omega\owns\eta\mapsto
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\KP1\RHB{.3}{^p}\in\lbb R_+ $} \inskipline{.3}0 | 3,485 | 362,273 | en |
train | 0.47.2 | For \mathss37{\vPi\aar 1\in\tvsps0(K)}, \,i.e.\ having \math{\vPi\aar 1} a
real or complex topological vector space with possibly non\kern0.37mm-\kern0.15mm Hausdorff
topology \mathss34{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\aar 1}, \,the exact construction of the space
$E=
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\aar 1)$ is given in
Constructions \ref{defi $L^p$}\,(\ref{simpL^p}) on page \pageref{simpL^p}
below. Here we informally explain the basic ideas
under the additional assumption that
$\vPi\aar 1$ is {\sl suitable\kern0.15mm} in the sense of
Definitions \ref{df suit} on page \pageref{df suit} below. Then
it suffices to consider one fixed
{\sl dominating\kern0.15mm} norm $\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ for $\vPi\aar 1\kern0.37mm$.
We consider functions
$x:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi$ such that on every set $A$ of finite measure, i.e.\
for $A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ it holds that outside some set $N$ of
measure zero, i.e.\ with $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ we have $x$
a pointwise limit of a sequence of simple functions, with convergence in the
sense of the topology $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$. In the case $p<\lower1.05mm\hbox{$^+$}\infty$ we then take
the subset of those $x$ such that the generally nonmeasurable
function \vskip.3mm\centerline{$
\Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x:{}^{}\Cal Omega\owns\eta\mapsto
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\KP1\RHB{.3}{^p}\in\lbb R_+ $} \inskipline{.3}0
is dominated by some integrable function
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm:{}^{}\Cal Omega\to[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] \,$. With the pointwise
vector operations from $\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ the set of these $x$ becomes a
vector substructure $X$ of $\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.37mm]_{vs} \,$.
Then we take $E=(\kern0.15mm X\kern0.15mm/\vsquotient N\aar 0\,,\kern0.07mm\scrmt T\,)$ when $
N\aar 0$ is the set of all $x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mms X$ such that for all
$u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ and
$A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ we have
$\int_{\,A\,}u\circ x\rmdss11\mu=0 \,$. Here
$\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ is the set of all
continuous linear maps $\vPi\to\bosy K\kern0.37mm$. Furthermore, we take the
topology $\scrmt T$ so that
a filter of zero neighbourhoods is formed by the sets \vskip.3mm\centerline{$
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mms(\kern0.15mm X\kern0.15mm/\vsquotient N\aar 0\kern0.15mm)
\capss31\{\,\smb X:
\eexi{x\in\smb X}\,
\upint\kern0.37mm \Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\rmdss41\mu < \varepsilon\KPt9\} $} \inskipline{.3}0
for $\varepsilon\in\rbb R^+$. Here we have the upper integral
of the
not\kern0.37mm-\kern0.15mm necessarily measurable function
$\Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x$ that is defined
as the infimum of the set of all
$\int_{\KP{1.1}{}^{}\Cal Omega\,}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu$ with $\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm$ as above.
For $p=\lower1.05mm\hbox{$^+$}\infty$ we take the \q{obvious} modification.
The space \math{E\ar 1=
\mvsLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\aar 1)
} is constructed otherwise similarly except that
we instead require the functions $x$ to be such that
$u\circ x\KP1|\KP1(\kern0.15mm A\kern0.15mm\setminus N\kern0.37mm)$ is measurable, that is,
for every
$A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ we require existence of
some $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that for all
$u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ it holds that
$u\circ x\KP1|\KP1(\kern0.15mm A\kern0.15mm\setminus N\kern0.37mm)$ is a measurable real or
complex valued function on $A\kern0.15mm\setminus N$. Then every vector of $E$ is
contained in some vector of \mathss34{E\ar 1}, \,but we need not have \mathss34{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\ar 1 }. Note above that \mathss39{ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E =
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mms(\kern0.15mm X\kern0.15mm/\vsquotient N\aar 0\kern0.07mm) }, \,and that \mathss03{
X\kern0.15mm/\vsquotient N\aar 0 } is the quotient vector space structure of \math{X}
by the linear subspace \mathss36{N\aar 0}.
Having now informally explained the general construction of our generalized
Bochner spaces, we note that if \math{\vPi} is \erm Banachable, then \math{
\vPi\dualsigma0} is its weak dual space, and that \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\kern0.15mm,\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm) } is a Banach space for any compatible norm \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \mathss31{\vPi}. At least for \rsigma6finite positive measures \math{
\mu} then \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\,,\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 2\kern0.15mm) } is a
classical Bochner space when \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aar 2} is defined by \math{ \smb X
\mapsto\big(\kern0.15mm\int_{\KP{1.1}{}^{}\Cal Omega\,}\Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x
\rmdss11\mu\kern0.37mm)\KP1\RHB{.2}{^p}{^{^{-1}}} } for any \mathss31{x\in\smb X}.
The appearing \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is the function \math{ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1 \owns \smb Y
\mapsto\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss02\smb Y} with \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss02\smb Y } given by \vskip.3mm\centerline{$
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F \owns \smb X \mapsto \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu $} \inskipline{.3}0
for any \math{x\in\smb X} and \mathss30{y\in\smb Y}. Here \math{y\,.\KPt8 x}
is the function \vskip.3mm\centerline{$
{}^{}\Cal Omega\owns\eta\mapsto y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) =
(\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) \in \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K \in
\{\ssbb97 R,\kern0.07mm\ssbb08 C\} \KP1 $.} \inskipline{.3}0 | 3,940 | 362,273 | en |
train | 0.47.3 | Having now informally explained the general construction of our generalized
Bochner spaces, we note that if \math{\vPi} is \erm Banachable, then \math{
\vPi\dualsigma0} is its weak dual space, and that \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\kern0.15mm,\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm) } is a Banach space for any compatible norm \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \mathss31{\vPi}. At least for \rsigma6finite positive measures \math{
\mu} then \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\,,\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 2\kern0.15mm) } is a
classical Bochner space when \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aar 2} is defined by \math{ \smb X
\mapsto\big(\kern0.15mm\int_{\KP{1.1}{}^{}\Cal Omega\,}\Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x
\rmdss11\mu\kern0.37mm)\KP1\RHB{.2}{^p}{^{^{-1}}} } for any \mathss31{x\in\smb X}.
The appearing \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is the function \math{ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1 \owns \smb Y
\mapsto\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss02\smb Y} with \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss02\smb Y } given by \vskip.3mm\centerline{$
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F \owns \smb X \mapsto \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu $} \inskipline{.3}0
for any \math{x\in\smb X} and \mathss30{y\in\smb Y}. Here \math{y\,.\KPt8 x}
is the function \vskip.3mm\centerline{$
{}^{}\Cal Omega\owns\eta\mapsto y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) =
(\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) \in \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K \in
\{\ssbb97 R,\kern0.07mm\ssbb08 C\} \KP1 $.} \inskipline{.3}0
The message of Theorem \nfss A\,\ref{main Th} is then that \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm \in
\Lis(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\dlbetss10\kern0.15mm) } holds, i.e.\ that \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is
a linear homeomorphism \math{F\aar 1\to F\dlbetss10} where \math{F\dlbetss10}
is the normable, hence \erm Banachable strong dual of \math{F} with \mathss38{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm(\kern0.15mm F\dlbetss10\kern0.15mm)=\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }.
For
the spaces appearing in (3) and \linebreak
(4) note that we define \mathss39{
\mLrs03^p(\kern0.37mm\mu\kern0.15mm) = \mvLrs03^p(\kern0.37mm\mu\,,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) }. Below note that
in the usual man- ner we have \math{ p\,^* =
(\kern0.37mm 1 - p\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm\big){}^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1} } for \math{
1 < p < \lower1.05mm\hbox{$^+$}\infty} and \math{p\,^*=\lower1.05mm\hbox{$^+$}\infty} for \mathss34{p=1}, \,and also $
p\,^*=1\kern0.37mm$ in the case where \math{p=\lower1.05mm\hbox{$^+$}\infty} holds.
The last part in condition (4) means that
there is a continuous linear map
$c:\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm)\to
\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.15mm{}^{}\Cal Omega\kern0.15mm)$
such that $c\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X\in\smb X$ holds for all $
\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm) \, $.
Continuity here being equivalent to the property that
for some $\smb M\in\lbb R_+$ it holds that
for $x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm
\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm)$ and for all $
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+
$ there is $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
$
|\KP1 c\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|
\le\smb M\KP1|\KP1 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|$ holds for all $\eta\in
A\kern0.15mm\setminus N$, our condition is
weaker than the requirement (b) in
\cite[Theorem 8.18.2\kern0.37mm, p.\ 588]{Edw} that \math{
\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm) } can be \q{lifted}.
\begin{remarks}\label{Rem Rad-deco}
At first sight, it may seem that Theorem \nfss A\,\ref{main Th} is less
general than the results contained in
\cite[Theorems 8.18.2\kern0.37mm, 8.18.3\kern0.37mm, pp.\ 588\kern0.37mm, 590]{Edw} when \math{p=1}
since in Edwards' presentation there is stated no assumption on any kind of
\q{decompos- ability}. However, one should note that in \cite{Edw} one
considers only positive measures that are {\sl positive \esl Radonian\kern0.15mm} in
the sense of \kern0.15mm Definitions \ref{df top deco}\,(4) on page \pageref{df pos Radon}
below, and that by Proposition \ref{Propo top-deco} these are \q{automatically}
almost decomposable. See also \cite[Proposition 4.14.9\kern0.37mm, p.\ 229]{Edw}\,.
We also remark the main ideas of the proof of \kern0.15mm Theorem \nfss A\,\ref{main Th}
are essentially, at least implicitly, contained in \cite[pp.\ 573\,--\,607]{Edw}
although it is not quite straight- forward to see the exact details from the
presentation there.
Note that in \cite{Edw} positive measures are obtained from positive linear
functionals in the vector spaces \math{
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm C\kern-0.3mm\sbi{\rm c}\kern0.37mm(\kern0.37mm\scrmt T\,) } of compactly supported continuous
functions for locally compact Hausdorff topologies \mathss30{\scrmt T}, \,
cf.\ \cite[4.3\kern0.37mm, pp.\ 177\,--\,179]{Edw}\,. Furthermore, in \cite{Edw}
measurability of functions is defined by the Lusin property which is
meaningless for general measures.
\end{remarks}
\begin{remark} | 3,443 | 362,273 | en |
train | 0.47.4 | The last part in condition (4) means that
there is a continuous linear map
$c:\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm)\to
\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.15mm{}^{}\Cal Omega\kern0.15mm)$
such that $c\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X\in\smb X$ holds for all $
\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm) \, $.
Continuity here being equivalent to the property that
for some $\smb M\in\lbb R_+$ it holds that
for $x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm
\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm)$ and for all $
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+
$ there is $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
$
|\KP1 c\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|
\le\smb M\KP1|\KP1 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|$ holds for all $\eta\in
A\kern0.15mm\setminus N$, our condition is
weaker than the requirement (b) in
\cite[Theorem 8.18.2\kern0.37mm, p.\ 588]{Edw} that \math{
\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm) } can be \q{lifted}.
\begin{remarks}\label{Rem Rad-deco}
At first sight, it may seem that Theorem \nfss A\,\ref{main Th} is less
general than the results contained in
\cite[Theorems 8.18.2\kern0.37mm, 8.18.3\kern0.37mm, pp.\ 588\kern0.37mm, 590]{Edw} when \math{p=1}
since in Edwards' presentation there is stated no assumption on any kind of
\q{decompos- ability}. However, one should note that in \cite{Edw} one
considers only positive measures that are {\sl positive \esl Radonian\kern0.15mm} in
the sense of \kern0.15mm Definitions \ref{df top deco}\,(4) on page \pageref{df pos Radon}
below, and that by Proposition \ref{Propo top-deco} these are \q{automatically}
almost decomposable. See also \cite[Proposition 4.14.9\kern0.37mm, p.\ 229]{Edw}\,.
We also remark the main ideas of the proof of \kern0.15mm Theorem \nfss A\,\ref{main Th}
are essentially, at least implicitly, contained in \cite[pp.\ 573\,--\,607]{Edw}
although it is not quite straight- forward to see the exact details from the
presentation there.
Note that in \cite{Edw} positive measures are obtained from positive linear
functionals in the vector spaces \math{
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm C\kern-0.3mm\sbi{\rm c}\kern0.37mm(\kern0.37mm\scrmt T\,) } of compactly supported continuous
functions for locally compact Hausdorff topologies \mathss30{\scrmt T}, \,
cf.\ \cite[4.3\kern0.37mm, pp.\ 177\,--\,179]{Edw}\,. Furthermore, in \cite{Edw}
measurability of functions is defined by the Lusin property which is
meaningless for general measures.
\end{remarks}
\begin{remark}
Using Theorem \nfss A\,\ref{main Th} one is able to prove
\cite[5.22\kern0.37mm, p.\ 27]{Am97} in the more general case where only separability
of the topology \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} is required instead of having \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm) } separable. Then for example in the case \math{
\vPi=\LLrs42^1(\ssbb44 I) } the strong dual of the Besov space \math{
\Besovrss600_q^{\emath s\kern0.15mm,\,p}\sbig(2\yi N\ssbb67 R,\kern0.07mm\vPi\kern0.37mm) } is seen
to be canonically represented by \linebreak
\ú$
\Besovrss300_{q\sast}^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}\emath s\kern0.15mm,\,p\sast}\kern-0.63mm \sbig(2\yi N\ssbb67 R,\kern0.07mm
\vPi\dlbetss01\kern0.15mm) \kern0.37mm $ when \math{s\in\mathbb R} and \math{1\le p<\lower1.05mm\hbox{$^+$}\infty }
and \math{1\le q<\lower1.05mm\hbox{$^+$}\infty} and \math{\smb N\in\mathbb N} hold. \linebreak
This is in
constrast with the case of Bessel potential spaces where the strong dual \linebreak
of \math{
\HBsrss606^{\emath s\kern0.15mm,\,p}\sbig(2\yi N\ssbb67 R,\kern0.07mm\vPi\kern0.37mm) } is
represented only by \mathss38{\HBsrss300^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}\emath s\kern0.15mm,\,p\sast}
\sbig(2\yi N\ssbb67 R,\kern0.07mm\vPi\dlsigss00\kern0.07mm) }. We hope to have the
opportunity to give the details of the proof in a future publication.
\end{remark}
We shortly review the {\font\≈=cmss10\≈it contents\kern0.37mm} which is organized according
to the scheme: \vskip.5mm {\newcommand\nrm[1]{$\null$\kern2.8mm{\font\≈=cmr9\≈#1\kern.4mm}}\newcommand\ntrm[2]{$\null$\kern2.8mm\ref{#1}\,{\font\≈=cmr8\≈#2\kern.4mm}}
\parindent6mm \inskipline07
1. Some special constructions \dotfill \ p.\KP{2.95} \pageref{Ss spec ctrs} \KP4 \inskipline07
2. Suitable locally convex spaces \dotfill \ p.\ \pageref{Ss suit lcs} \KP4
A \ Measurability and integration \dotfill \ p.\ \pageref{Sec A} \KP4 \inskipline07
1. Measurability of measure\kern0.37mm-\kern0.15mm vector maps \dotfill \ p.\ \pageref{Ss C1} \KP4 \inskipline07
2. Decomposable positive measures \dotfill \ p.\ \pageref{Ss decos} \KP4 \inskipline07
3. Integration of scalar functions \dotfill \ p.\ \pageref{Ss int scal} \KP4 \inskipline07
4. Pettis integration of vector functions \dotfill \ p.\ \pageref{Ss Pettis} \KP4
B \ Generalized Bochner spaces \dotfill \ p.\ \pageref{Sec B} \KP4
C \ Lifting and integral representations \dotfill \ p.\ \pageref{Sec C} \KP4 \inskipline07
1. Dunford\,--\,Pettis property of $\,\mLrs42^1(\kern0.37mm\mu\kern0.37mm)$ \dotfill \ p.\ \pageref{Ss Dun-Pet} \KP4 \inskipline07
2. Absolutely continuous vector measures \dotfill \ p.\ \pageref{Ss abs conti} \KP4
D \ Duality of Bochner spaces \dotfill \ p.\ \pageref{Sec D} \KP4
E \ Examples and open problems \dotfill \ p.\ \pageref{Sec E} \KP4 \inskipline0{9.2}
References \dotfill \ p.\ \pageref{Sec Bib} \KP4
\par} \vskip.5mm
In subsection 1 of this introductory section we give some special
constructions in order to be able to express certain matters concisely and
precisely at the same time. In 2 we give the basic definitions associated with
suitable spaces. We also establish some lemmas that are needed in the sequel.
In section A we present our approach to measurability and integration of
scalar and vector valued functions, or put more precisely,
{\sl mv\kern0.37mm-\kern0.15mm map\kern0.37mm}s. These are triplets \math{
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } where \math{\vPi} is a real or complex
topological vector space and \math{\mu} is a positive measure on some set \math{
{}^{}\Cal Omega} and \math{x:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } is a function.
In section B we first give the formal construction of our generalized
Lebesgue\,--\,Bochner spaces of equivalence classes \math{\smb X} of
measurable functions \math{x:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } when a positive measure \math{
\mu} on \math{{}^{}\Cal Omega} is given. Then we prove several results associated with
these spaces that are needed in the proof of our main theorem.
Section C contains several auxiliary results that are needed to prove that a
given continuous linear functional \math{ \smb U \kern-0.3mm :
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) = F \to \bosy K } can be represented by
some vector \math{\smb Y} of the space \math{F\aar 1} in the sense that for \math{
x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} and \math{y\in\smb Y} we have the equality \mathss36{
\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X = \int_{\KPp1.1{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu}.
At the beginning of section D we note how other assertions
of \kern0.15mm Theorem \nfss A\,\ref{main Th} except surjectivity of \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}
follow from results that have already been established in section B\kern0.37mm. Then
we prove the surjectivity in Lemmas \nfss A\,\ref{LeA(1)}$\,,\ldots\KPt8
$\nfss A\,\ref{final lemma} separately in the cases (1)$\,,\ldots\KPt8$(6)
with (5) and (6) being treated together in \nfss A\,\ref{final lemma}\kern0.37mm.
In section E we have collected examples to make more concrete some points of
the general theory. We also present some related open problems. \vskip.5mm
Above we already indicated that \math{\tvsps0(K)} is the class of all
topological vector spaces over \math{\bosy K} when \math{\bosy K} is a
topological field. We put \vskip.3mm\centerline{$
\roman{TVS}\kern0.4mmps0(K) = \tvsps0(K)\capss31\{\, E :
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm\text{ is a Hausdorff topology } \} \KP1$,} \inskipline{.3}0 | 3,961 | 362,273 | en |
train | 0.47.5 | 2. Decomposable positive measures \dotfill \ p.\ \pageref{Ss decos} \KP4 \inskipline07
3. Integration of scalar functions \dotfill \ p.\ \pageref{Ss int scal} \KP4 \inskipline07
4. Pettis integration of vector functions \dotfill \ p.\ \pageref{Ss Pettis} \KP4
B \ Generalized Bochner spaces \dotfill \ p.\ \pageref{Sec B} \KP4
C \ Lifting and integral representations \dotfill \ p.\ \pageref{Sec C} \KP4 \inskipline07
1. Dunford\,--\,Pettis property of $\,\mLrs42^1(\kern0.37mm\mu\kern0.37mm)$ \dotfill \ p.\ \pageref{Ss Dun-Pet} \KP4 \inskipline07
2. Absolutely continuous vector measures \dotfill \ p.\ \pageref{Ss abs conti} \KP4
D \ Duality of Bochner spaces \dotfill \ p.\ \pageref{Sec D} \KP4
E \ Examples and open problems \dotfill \ p.\ \pageref{Sec E} \KP4 \inskipline0{9.2}
References \dotfill \ p.\ \pageref{Sec Bib} \KP4
\par} \vskip.5mm
In subsection 1 of this introductory section we give some special
constructions in order to be able to express certain matters concisely and
precisely at the same time. In 2 we give the basic definitions associated with
suitable spaces. We also establish some lemmas that are needed in the sequel.
In section A we present our approach to measurability and integration of
scalar and vector valued functions, or put more precisely,
{\sl mv\kern0.37mm-\kern0.15mm map\kern0.37mm}s. These are triplets \math{
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } where \math{\vPi} is a real or complex
topological vector space and \math{\mu} is a positive measure on some set \math{
{}^{}\Cal Omega} and \math{x:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } is a function.
In section B we first give the formal construction of our generalized
Lebesgue\,--\,Bochner spaces of equivalence classes \math{\smb X} of
measurable functions \math{x:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } when a positive measure \math{
\mu} on \math{{}^{}\Cal Omega} is given. Then we prove several results associated with
these spaces that are needed in the proof of our main theorem.
Section C contains several auxiliary results that are needed to prove that a
given continuous linear functional \math{ \smb U \kern-0.3mm :
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) = F \to \bosy K } can be represented by
some vector \math{\smb Y} of the space \math{F\aar 1} in the sense that for \math{
x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} and \math{y\in\smb Y} we have the equality \mathss36{
\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X = \int_{\KPp1.1{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu}.
At the beginning of section D we note how other assertions
of \kern0.15mm Theorem \nfss A\,\ref{main Th} except surjectivity of \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}
follow from results that have already been established in section B\kern0.37mm. Then
we prove the surjectivity in Lemmas \nfss A\,\ref{LeA(1)}$\,,\ldots\KPt8
$\nfss A\,\ref{final lemma} separately in the cases (1)$\,,\ldots\KPt8$(6)
with (5) and (6) being treated together in \nfss A\,\ref{final lemma}\kern0.37mm.
In section E we have collected examples to make more concrete some points of
the general theory. We also present some related open problems. \vskip.5mm
Above we already indicated that \math{\tvsps0(K)} is the class of all
topological vector spaces over \math{\bosy K} when \math{\bosy K} is a
topological field. We put \vskip.3mm\centerline{$
\roman{TVS}\kern0.4mmps0(K) = \tvsps0(K)\capss31\{\, E :
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm\text{ is a Hausdorff topology } \} \KP1$,} \inskipline{.3}0
and we let \math{\roman{LCS}\kern0.4mmps0(K)} be the subclass of \math{\roman{TVS}\kern0.4mmps0(K)} formed by
the locally convex spaces. For \math{E\in\tvsps0(K)} we have \math{\bouSet E}
the set of all bounded sets in \mathss35{E}, also cal- led the {\sl von Neumann
bornology\kern0.15mm} of \mathss35{E}. For \math{E\,,\kern0.15mm F\in\tvsps0(K) } we let \math{
E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} mean \linebreak
that the identity \math{\kern.3mm\roman{id}\kern.7mmv F} is a continuous linear map \mathss35{
F\to E}.
If \math{E} is a real or complex topological vector space, then \math{
\Cal S\sbi{\kern0.15mm\emath r\,}E } and \math{\BSnorm E} and \linebreak
\ú$\Bqnorm E\kern0.37mm$ are the
sets of continuous \mathss35{r}--\,seminorms, bounded seminorms and bounded
quasi\kern0.37mm-\kern0.37mm seminorms, respectively, the formal constructions being given
in (1)$\,,\ldots\,$(3) below. We also put \math{ \Cal S_{_N}\kern0.15mm E =
\Cal S\LHB{.3}{_{\,1\,}} E} thus getting the set of continuous seminorms.
Note the implication \math{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm \in \Cal S\sbi{\kern0.15mm\emath r\,}E \impss33
0 < r \le 1 } and that a quasi\kern0.37mm-\kern0.37mm seminorm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} being
{\sl bounded\kern0.37mm} means that \math{ \sup \KP1 (\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm B\kern0.37mm) <
\lower1.05mm\hbox{$^+$}\infty } holds for every \mathss35{B\in\bouSet E}. We generally have \mathss35{
\Cal S_{_N}\kern0.15mm E\subseteq\BSnorm E}, \,and the converse inclusion holds if \math{E} is
normable.
\begin{enumerate}\begin{myLeftskip}{-2}{.5}{.3}
\item \ $\Cal S\sbi{\kern0.15mm\emath r\,}E = \kern0.15mm^{\svecs E}\KP1\lbb R_+\kern-0.2mm\cap\kern0.37mm
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm : 0 < r \le 1\kern0.37mm\text{ and }\kern0.37mm \label{df r-semin E}
\aall{t\kern0.37mm,\kern0.15mm x\kern0.37mm,\kern0.15mm y\kern0.37mm,\kern0.15mm z}\, $ \newskline{22}
$[\KP{1.4}(\kern0.37mm t\kern0.37mm,\kern0.07mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y = |\,t\,|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x\kern0.37mm)\KP{1.4}] \kern0.37mm$ and \newskline{22}
$[\KP{1.4}(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm)\in\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm E\impss33
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 z\kern0.37mm)\RHB{.2}{\KPt8^{\emath r}} \kern-0.3mm \le \kern0.15mm
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x\kern0.37mm)\RHB{.2}{\KPt8^{\emath r}\kern-0.3mm} +
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y\kern0.37mm)\RHB{.2}{\KPt8^{\emath r}} \KPp1.4\big]$ \newfline
and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} is continuous \mathss39{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\to\nsTbb_R \, \} }, \KP{10}
\item \ $\Bqnorm E = \kern0.15mm^{\svecs E}\KP1\lbb R_+\kern-0.2mm\cap\kern0.37mm \{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm : \label{defi bqnor E}
\eexi{\smb A}\,\aall{t\kern0.37mm,\kern0.15mm x\kern0.37mm,\kern0.15mm y\kern0.37mm,\kern0.15mm z}\,
\smb A\in\rbb R^+\kern0.15mm$ and \newskline{22}
$[\KP{1.4}(\kern0.37mm t\kern0.37mm,\kern0.07mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y = |\,t\,|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x\kern0.37mm)\KP{1.4}] \kern0.37mm$ and \newskline{22} | 3,805 | 362,273 | en |
train | 0.47.6 | \begin{enumerate}\begin{myLeftskip}{-2}{.5}{.3}
\item \ $\Cal S\sbi{\kern0.15mm\emath r\,}E = \kern0.15mm^{\svecs E}\KP1\lbb R_+\kern-0.2mm\cap\kern0.37mm
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm : 0 < r \le 1\kern0.37mm\text{ and }\kern0.37mm \label{df r-semin E}
\aall{t\kern0.37mm,\kern0.15mm x\kern0.37mm,\kern0.15mm y\kern0.37mm,\kern0.15mm z}\, $ \newskline{22}
$[\KP{1.4}(\kern0.37mm t\kern0.37mm,\kern0.07mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y = |\,t\,|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x\kern0.37mm)\KP{1.4}] \kern0.37mm$ and \newskline{22}
$[\KP{1.4}(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm)\in\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm E\impss33
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 z\kern0.37mm)\RHB{.2}{\KPt8^{\emath r}} \kern-0.3mm \le \kern0.15mm
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x\kern0.37mm)\RHB{.2}{\KPt8^{\emath r}\kern-0.3mm} +
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y\kern0.37mm)\RHB{.2}{\KPt8^{\emath r}} \KPp1.4\big]$ \newfline
and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} is continuous \mathss39{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\to\nsTbb_R \, \} }, \KP{10}
\item \ $\Bqnorm E = \kern0.15mm^{\svecs E}\KP1\lbb R_+\kern-0.2mm\cap\kern0.37mm \{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm : \label{defi bqnor E}
\eexi{\smb A}\,\aall{t\kern0.37mm,\kern0.15mm x\kern0.37mm,\kern0.15mm y\kern0.37mm,\kern0.15mm z}\,
\smb A\in\rbb R^+\kern0.15mm$ and \newskline{22}
$[\KP{1.4}(\kern0.37mm t\kern0.37mm,\kern0.07mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y = |\,t\,|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x\kern0.37mm)\KP{1.4}] \kern0.37mm$ and \newskline{22}
$[\KP{1.4}(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm)\in\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm E\impss33 \label{defi bqnor E p}
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 z \le \smb A\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x + \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y\kern0.37mm)
\KP{1.4}]$ \newfline
and $\kern0.37mm \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\KPt7\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.07mm\bouSet E\subseteq\bouSet\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R \KP1 \} \KP1 $, \KP{10}
\item \ $\BSnorm E = \Bqnorm E\capss30\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm:\aall{x\kern0.37mm,\kern0.15mm y\kern0.37mm,\kern0.15mm z}\, \label{defi bsnor E}
(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm)\in\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm E$ \newfline
$\impss03 \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 z \le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x + \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y\KPt9\}\KP1$. \KP{10}
\end{myLeftskip}\end{enumerate}
\insubsubhead Some special constructions \label{Ss spec ctrs}
We are working within a Kelley\,--\,Morse\,--\,G\"odel\,--\,Bernays\,--\,von
Neumann type approach to set theory, like for example the one introduced in
\cite[pp.\ 250\,--\,281]{Ky}\,. Then with \math{ x\kern0.37mm\lower1mm\hbox{$^{^+}$} =
x\cupss21\{\kern0.37mm x\kern0.37mm\} } putting \mathss38{ \mathbb No = \kern0.37mm
\bigcap\KPt8\{\,N\kern-0.3mm:\emptyset\in N\kern0.37mm\text{ and }\kern0.37mm\aall{k\in N}\,
k\kern0.37mm\lower1mm\hbox{$^{^+}$}\in N\KPt9\} }, \,we may call \math{\mathbb No} the set of {\sl natural
number\kern0.15mm}s. It equals the set of finite cardinals, as well as the set of
finite ordinals. Let \math{\infty=\mathbb No} and \mathss38{ \mathbb N =
\mathbb No\kern-0.3mm\setminus\{\kern0.37mm\emptyset\kern0.37mm\} }.
We assume that the set \math{\mathbb H} of {\sl quaternion\kern0.15mm}s is constructed
in a certain manner so that we have \math{\mathbb H\subseteq
\ovbbR\ar 1\kern-0.3mm\times\ovbbR\ar 1\kern-0.3mm\times(\kern0.37mm\ovbbR\ar 1\kern-0.3mm\times\ovbbR\ar 1) }
for some set \math{\ovbbR\ar 1} with \vskip.3mm\centerline{$
\ovbbR\ar 1 \subseteq \Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.37mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.37mm\mathbb No\kern-0.2mm\times\mathbb N\kern0.37mm)\times
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.37mm\mathbb No\kern-0.2mm\times\mathbb N\kern0.37mm)) $} \inskipline{.3}0
where the {\sl power class\kern0.15mm} \math{\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A} of \math{A} is defined in
Definitions \ref{misc defs}\,(14) below. Then for some set \math{ 0\ar 1 \in
\ovbbR\ar 1} we have \math{\mathbb R \subseteq \ovbbR\ar 1\kern-.2mm\times\kern-.2mm\{\,0\ar 1\kern-0.2mm\}\kern-0.2mm
\times\sbig(2\{\,0\ar 1\kern-0.2mm\}\kern-.2mm\times\kern-.2mm\{\,0\ar 1\kern-0.2mm\}\kern0.15mm\sbig)0 } and \inskipline{.2}{41.7}
$\mathbb C \subseteq \ovbbR\ar 1\kern-0.3mm\times \ovbbR\ar 1\kern-0.3mm\times\sbig(2
\{\,0\ar 1\kern-0.2mm\}\kern-.2mm\times\kern-.2mm\{\,0\ar 1\kern-0.2mm\}\kern0.15mm\sbig)0 \KP1 $. \inskipline{.2}0 | 3,368 | 362,273 | en |
train | 0.47.7 | \item \ $\BSnorm E = \Bqnorm E\capss30\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm:\aall{x\kern0.37mm,\kern0.15mm y\kern0.37mm,\kern0.15mm z}\, \label{defi bsnor E}
(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm)\in\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm E$ \newfline
$\impss03 \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 z \le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x + \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y\KPt9\}\KP1$. \KP{10}
\end{myLeftskip}\end{enumerate}
\insubsubhead Some special constructions \label{Ss spec ctrs}
We are working within a Kelley\,--\,Morse\,--\,G\"odel\,--\,Bernays\,--\,von
Neumann type approach to set theory, like for example the one introduced in
\cite[pp.\ 250\,--\,281]{Ky}\,. Then with \math{ x\kern0.37mm\lower1mm\hbox{$^{^+}$} =
x\cupss21\{\kern0.37mm x\kern0.37mm\} } putting \mathss38{ \mathbb No = \kern0.37mm
\bigcap\KPt8\{\,N\kern-0.3mm:\emptyset\in N\kern0.37mm\text{ and }\kern0.37mm\aall{k\in N}\,
k\kern0.37mm\lower1mm\hbox{$^{^+}$}\in N\KPt9\} }, \,we may call \math{\mathbb No} the set of {\sl natural
number\kern0.15mm}s. It equals the set of finite cardinals, as well as the set of
finite ordinals. Let \math{\infty=\mathbb No} and \mathss38{ \mathbb N =
\mathbb No\kern-0.3mm\setminus\{\kern0.37mm\emptyset\kern0.37mm\} }.
We assume that the set \math{\mathbb H} of {\sl quaternion\kern0.15mm}s is constructed
in a certain manner so that we have \math{\mathbb H\subseteq
\ovbbR\ar 1\kern-0.3mm\times\ovbbR\ar 1\kern-0.3mm\times(\kern0.37mm\ovbbR\ar 1\kern-0.3mm\times\ovbbR\ar 1) }
for some set \math{\ovbbR\ar 1} with \vskip.3mm\centerline{$
\ovbbR\ar 1 \subseteq \Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.37mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.37mm\mathbb No\kern-0.2mm\times\mathbb N\kern0.37mm)\times
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.37mm\mathbb No\kern-0.2mm\times\mathbb N\kern0.37mm)) $} \inskipline{.3}0
where the {\sl power class\kern0.15mm} \math{\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A} of \math{A} is defined in
Definitions \ref{misc defs}\,(14) below. Then for some set \math{ 0\ar 1 \in
\ovbbR\ar 1} we have \math{\mathbb R \subseteq \ovbbR\ar 1\kern-.2mm\times\kern-.2mm\{\,0\ar 1\kern-0.2mm\}\kern-0.2mm
\times\sbig(2\{\,0\ar 1\kern-0.2mm\}\kern-.2mm\times\kern-.2mm\{\,0\ar 1\kern-0.2mm\}\kern0.15mm\sbig)0 } and \inskipline{.2}{41.7}
$\mathbb C \subseteq \ovbbR\ar 1\kern-0.3mm\times \ovbbR\ar 1\kern-0.3mm\times\sbig(2
\{\,0\ar 1\kern-0.2mm\}\kern-.2mm\times\kern-.2mm\{\,0\ar 1\kern-0.2mm\}\kern0.15mm\sbig)0 \KP1 $. \inskipline{.2}0
The definitions of the sets \math{\mathbb Z} and \math{\lbb Z_+} of
{\sl integers\kern0.15mm} and {\sl nonnegative\kern0.15mm} integers, respectively, being given
in \ref{misc defs}\,(5) and \ref{misc defs}\,(7) below, we have a bijection \math{
\mathbb No\to\lbb Z_+} given by \mathss03{i\mapsto n=i\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}} with inverse \math{
n\mapsto i=n\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}} and now for example \mathss30{ i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056} =
(\kern0.37mm i + 1\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.07mm)\kern0.07mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056} = n + 1 } and \math{
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern-0.2mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056} = (\kern0.37mm i + 2\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.07mm)\kern0.07mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}=n + 2 } and \mathss37{
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1} = (\kern0.37mm n + 1\kern0.37mm)\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1} }. Also \math{
\emptyset = 0\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}} holds.
Having \mathss38{ \ovbbR = [\kern0.15mm\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\lower1.05mm\hbox{$^+$}\infty\KP1] = \{\,t :
\kern.2mm\lower1.05mm\hbox{$^-$}infty\le t\le\lower1.05mm\hbox{$^+$}\infty\KPt9\} }, \,we assume the formal definitions
having been arranged so that for all \math{u\kern0.37mm,\kern0.15mm v} we have \math{u\le v}
if{}f \math{\kern.2mm\lower1.05mm\hbox{$^-$}infty\le u\le v\le\lower1.05mm\hbox{$^+$}\infty} or
\math{u\kern0.37mm,\kern0.15mm v} are functions with
\math{u\cupss22 v\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} v\times\ovbbR } and \mathss38{{{}^{}{\rm dom}\,{}_{{}^{}}} u\subseteq
\{\,\eta:u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KPt8\} }. Hence if \math{u} and \math{v}
are extended real valued functions, then \math{u\le v} means that
we have \math{{{}^{}{\rm dom}\,{}_{{}^{}}} u\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} v} and that
\math{u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta} holds for all
\mathss34{\eta\in{{}^{}{\rm dom}\,{}_{{}^{}}} u}.
Furthermore \math{\emptyset\le v} is equivalent to having \math{v} a
function with \mathss38{{}^{}{\rm rng}\,{}_{{}^{}} v\subseteq\ovbbR}.
In order to specify some set theoretic notation already utilized above that
also has largely been explained in \cite[pp.\ 4\,--\,8]{HiDim} and
\cite[pp.\ 4\,--\,9]{SeBGN}\,, ending on page \pageref{END set th extract}
below, we next present an extract from \cite{Hif}\,.
We assume that the {\sl intuitive class\kern0.15mm} of all {\sl variable symbol\kern0.15mm}s
of our set theory is implicitly intuitively well\kern0.37mm-\kern0.37mm ordered so that it
makes sense to speak of the first variable (\kern0.15mm symbol\kern0.37mm) not possessing
some property.
\begin{def:al schemas}[set notation]\label{defi {F:...:P}}
Let \math{\mfrk F} be any term and \math{\mfrk P} a formula and $\afr x_1\kern0.15mm,
\ldots\,\mfrk x\kern0.37mm\ai k\kern0.37mm,\kern0.15mm\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l\kern0.37mm$
distinct variable symbols such that \math{\afr x_1\kern0.15mm,\ldots\,\mfrk x\kern0.37mm\ai k}
are precisely the variable symbols which have a free occurrence both in \math{
\mfrk F} and \math{\mfrak P} and are not in the list \math{\afr y_1\kern0.15mm,\ldots\,
\mfrk y\kern0.37mm\ai l}. Also let \math{\mfrk x} be the first variable symbol not
occurring free in \math{\mfrk F} or \math{\mfrk P}. Then we let \math{
\{\,\mfrk F:\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l\kern-0.63mm:\mfrk P\,\} =
\{\,\mfrk x:\exi{\afr x_1\kern0.15mm,\ldots\,\mfrk x\kern0.37mm\ai k}\,\mfrk x=\mfrk F\kern0.37mm$
and $\kern0.37mm\mfrk P\,\}\yxbtext{15}1b \kern0.15mm}.
In the case where \math{
\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l} is an empty list, we further let \math{
\{\,\mfrk F:\mfrk P\,\} = \{\,\mfrk F:\ :\mfrk P\,\} \kern0.15mm}.
The variable symbols which are free in the term \math{
\{\,\mfrk F:\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l\kern-0.63mm:\mfrk P\,\}} are (\kern0.37mm by
re- cursive definition\kern0.15mm) exactly those which are free either in \math{
\mfrk F} or \math{\mfrk P}, and are not in the list \math{\afr x_1\kern0.15mm,\ldots\,
\mfrk x\kern0.37mm\ai k}. The free variables of \math{\{\,\mfrk F:\mfrk P\,\}} are
precisely those which are free in \math{\mfrk F} or \math{\mfrk P} but not in
both of them.
\end{def:al schemas}
The above schemata, which we introduced to overcome the notational problem
presented in \cite[4 Notes, pp.\ 5\,--\,6]{Ky}\kern0.37mm, only provide reduction of
\PouN$\kern0.37mm\{\KPt8\mfrk F:\afr y_1\kern0.15mm,\ldots\,\mfrak y\kern0.37mm\ai l\kern-0.63mm:\mfrak P\,\}$
and \math{\{\KPt8\mfrk F:\mfrak P\,\} } to \math{
\{\KPt8\mfrk x:\mfrk Q\,\}\yxbtext{15}1b \kern0.15mm}. In order to be able to prove
something nontrivial about $\{\KPt8\mfrk x:\mfrk Q\,\}\yxbtext{15}1b \,$, we
need some {\font\≈=cmss10\≈it axioms\kern0.37mm}. As such, we accept all the formulas
\begin{enumerate}\begin{myLeftskip}{-4}{.3}{.1}
\itemb0_ax $u=v\equivss22\aall x\,x\in u\equivss22 x\in v\,$, \label{ax of extent} | 4,054 | 362,273 | en |
train | 0.47.8 | In order to specify some set theoretic notation already utilized above that
also has largely been explained in \cite[pp.\ 4\,--\,8]{HiDim} and
\cite[pp.\ 4\,--\,9]{SeBGN}\,, ending on page \pageref{END set th extract}
below, we next present an extract from \cite{Hif}\,.
We assume that the {\sl intuitive class\kern0.15mm} of all {\sl variable symbol\kern0.15mm}s
of our set theory is implicitly intuitively well\kern0.37mm-\kern0.37mm ordered so that it
makes sense to speak of the first variable (\kern0.15mm symbol\kern0.37mm) not possessing
some property.
\begin{def:al schemas}[set notation]\label{defi {F:...:P}}
Let \math{\mfrk F} be any term and \math{\mfrk P} a formula and $\afr x_1\kern0.15mm,
\ldots\,\mfrk x\kern0.37mm\ai k\kern0.37mm,\kern0.15mm\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l\kern0.37mm$
distinct variable symbols such that \math{\afr x_1\kern0.15mm,\ldots\,\mfrk x\kern0.37mm\ai k}
are precisely the variable symbols which have a free occurrence both in \math{
\mfrk F} and \math{\mfrak P} and are not in the list \math{\afr y_1\kern0.15mm,\ldots\,
\mfrk y\kern0.37mm\ai l}. Also let \math{\mfrk x} be the first variable symbol not
occurring free in \math{\mfrk F} or \math{\mfrk P}. Then we let \math{
\{\,\mfrk F:\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l\kern-0.63mm:\mfrk P\,\} =
\{\,\mfrk x:\exi{\afr x_1\kern0.15mm,\ldots\,\mfrk x\kern0.37mm\ai k}\,\mfrk x=\mfrk F\kern0.37mm$
and $\kern0.37mm\mfrk P\,\}\yxbtext{15}1b \kern0.15mm}.
In the case where \math{
\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l} is an empty list, we further let \math{
\{\,\mfrk F:\mfrk P\,\} = \{\,\mfrk F:\ :\mfrk P\,\} \kern0.15mm}.
The variable symbols which are free in the term \math{
\{\,\mfrk F:\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l\kern-0.63mm:\mfrk P\,\}} are (\kern0.37mm by
re- cursive definition\kern0.15mm) exactly those which are free either in \math{
\mfrk F} or \math{\mfrk P}, and are not in the list \math{\afr x_1\kern0.15mm,\ldots\,
\mfrk x\kern0.37mm\ai k}. The free variables of \math{\{\,\mfrk F:\mfrk P\,\}} are
precisely those which are free in \math{\mfrk F} or \math{\mfrk P} but not in
both of them.
\end{def:al schemas}
The above schemata, which we introduced to overcome the notational problem
presented in \cite[4 Notes, pp.\ 5\,--\,6]{Ky}\kern0.37mm, only provide reduction of
\PouN$\kern0.37mm\{\KPt8\mfrk F:\afr y_1\kern0.15mm,\ldots\,\mfrak y\kern0.37mm\ai l\kern-0.63mm:\mfrak P\,\}$
and \math{\{\KPt8\mfrk F:\mfrak P\,\} } to \math{
\{\KPt8\mfrk x:\mfrk Q\,\}\yxbtext{15}1b \kern0.15mm}. In order to be able to prove
something nontrivial about $\{\KPt8\mfrk x:\mfrk Q\,\}\yxbtext{15}1b \,$, we
need some {\font\≈=cmss10\≈it axioms\kern0.37mm}. As such, we accept all the formulas
\begin{enumerate}\begin{myLeftskip}{-4}{.3}{.1}
\itemb0_ax $u=v\equivss22\aall x\,x\in u\equivss22 x\in v\,$, \label{ax of extent}
\itemb0_ax $u\in v\impss22\eexi{w\kern0.37mm,z}\,w\in z\kern0.37mm$ and $\,
\aall x\,x\subseteq u\impss22 x\in w\kern0.37mm$, \label{ax of subsets}
\itemb0_ax $x\in u\kern0.37mm$ and $\kern0.37mm y\in v\impss22\eexi w\,
x\subseteq w\kern0.37mm$ and $\kern0.37mm y\subseteq w\kern0.37mm$, \label{ax of union}
\itemb0_ax $u\in z\kern0.37mm$ and $\,[\KP{1.4}\aall{x\kern0.37mm,y\kern0.37mm,z}\,
(\kern0.37mm x\kern0.37mm,y\kern0.15mm)\kern0.37mm,(\kern0.37mm x\kern0.37mm,z\kern0.15mm)\in f \impss22
y=z \KP{1.4} ] \impss22 \eexi{v\kern0.37mm,w}$ \newfline
$v\in w\kern0.37mm$ and $\,\aall y\,y\in v\equivss22\eexi x\,x\in u\kern0.37mm$
and $\kern0.37mm(\kern0.37mm x\kern0.37mm,y\kern0.15mm)\in f\kern0.15mm$, \KP{17.9} \label{ax of substi}
\itemb0_ax $z\in w\impss22\eexi{u\kern0.37mm,v}\,u\in v\kern0.37mm$ and $\,\aall x\,
x\in u\equivss22\eexi y\,x\in y\in z\,$, \label{ax of amalg}
\itemb0_ax $v\in u\impss22\eexi x\,x\in u\kern0.37mm$ and not $\eexi z\,
z\in x\kern0.37mm$ and $\kern0.37mm z\in u\,$, \label{ax of regularity}
\itemb0_ax $\eexi{e\kern0.37mm,\kern0.07mm N\kern0.15mm,\kern0.07mm S}\,e\in N\in S\kern0.37mm$ and \,[\ not $
\eexi x\,x\in e \KP{1.4} ]\kern0.37mm$ and $\,\aall{n\kern0.37mm,\kern0.07mm m}$ \newfline
$n\in N\kern0.37mm$ and $\,[\KP{1.4}\aall x\,x\in m\equivss22
x\in n\kern0.37mm$ or $\kern0.37mm x=n \KP{1.4} ]\impss22 m\in N\kern0.37mm$, \KP{17.9} \label{ax of infin}
\itemb0_ax $\eexi C\,[ \KP{1.4} \aall{x\kern0.37mm,z\kern0.37mm,u}\,(\kern0.15mm u\kern0.37mm,x\kern0.15mm)\kern0.37mm,
(\kern0.15mm u\kern0.37mm,z\kern0.15mm)\in C\impss22 x=z\in u \KP{1.4} ]\kern0.37mm$ and \newfline
$\aall{z\kern0.37mm,u\kern0.37mm,w}\,z\in u\in w\impss22\eexi x\,
(\kern0.15mm u\kern0.37mm,x\kern0.15mm)\in C\kern0.37mm$, \KP{17.9} \label{ax of choice}
\end{myLeftskip}\end{enumerate}
\noindent and also all the formulas (\kern0.15mm s\kern0.15mm) given in the next
\begin{axiom schema}[classification]\label{class axi}
Let \math{\mfrk x} be any variable symbol and \math{\mfrk P} any formula. Let
\math{\mfrk y} be the first variable symbol distinct from \math{\mfrk x} and
not occurring free in \math{\mfrk P}. Then we accept as an axiom the formula
(\kern0.15mm s\kern0.15mm) \ $\mfrk x\in\{\KPt8\mfrk x:\mfrk P\,\}\yxbtext{15}1b\equivss22
\eexi{\mfrk y}\,\mfrk x\in\mfrk y\kern0.37mm$ and $\kern0.37mm\mfrk P\,$.
\end{axiom schema}
Above (\ref{ax of choice})$\ar{ax}$ is the {\sl global axiom of choice\kern0.15mm} and
(\ref{ax of infin})$\ar{ax}$ is the {\sl axiom of infinity\kern0.15mm}.
\begin{remark}\label{rem about class sch}
Among others, we accept as logical axioms the formulas \inskipline{.3}3
(1)$\ar{az}\,$ \ $\mfrk P\impss22\aall{\mfrk x}\,\mfrk P\,$, \KP{20}
(2)$\ar{az}\,$ \ $[ \KP{1.4} \aall{\mfrk x}\,\mfrk Q \KP{1.4} ]\impss22
\mfrk Q\,(\kern0.37mm\mfrk x\kern-0.3mm\lleftarrow\kern-0.3mm\mfrak F\kern0.37mm) \,$, \inskipline{.3}0
when \math{\mfrk x} is any variable symbol and \math{\mfrk F} is any term and
\math{\mfrk P\kern0.37mm,\mfrk Q} are any formulas such that for any variable symbol \math{
\mfrk y} having a free occurrence in \math{\mfrk F} the bound (\kern0.37mm i.e.\
non\kern0.37mm-\kern0.37mm free\kern0.15mm) occurrences of \math{\mfrk y} in \math{\mfrk Q} and \math{
\mfrk Q\,(\kern0.37mm\mfrk x\kern-0.3mm\leftarrow\kern-0.3mm\mfrak y\kern0.37mm)} are the same. Having these
logical axioms, we could give Axiom schema \ref{class axi} above a simpler
formulation than has the corresponding \cite[\erm{II}\kern0.37mm, p.\ 253]{Ky} which
in our notation would (\kern0.15mm as already a bit corrected\kern0.37mm) read as follows.
For any variable symbols \math{\mfrk x\kern0.37mm,\kern0.15mm\mfrk y\kern0.37mm,\kern0.15mm\mfrk z} and for
any formula \mathss30{\mfrk P} such that \math{\mfrk y} is the first one
distinct from \math{\mfrk x} and \mathss34{\mfrk z}, \,and not occurring free
in \mathss34{\mfrk P}, we accept as an axiom the formula \inskipline{.3}3
(t) \ $\aall{\mfrk z}\,\mfrk z\in\{\,\mfrk x:\mfrk P\,\}\yxbtext{15}1b\equivss22
\eexi{\mfrk y}\,\mfrk z\in\mfrk y\kern0.37mm$ and $\kern0.37mm
\mfrk P\,(\kern0.37mm\mfrk x\kern-0.3mm\leftarrow\kern-0.3mm\mfrak z\kern0.37mm)\,$. \inskipline{.3}0
However, this would make the system contradictory as shown in Example \ref{exa about Kelley's class sch}
below. One should put the additional restriction that the bound occurrences of \math{
\mfrk z} in $\kern0.37mm\mfrk P$ \linebreak
and \math{
\mfrk P\,(\kern0.37mm\mfrk x\kern-0.3mm\leftarrow\kern-0.3mm\mfrk z\kern0.37mm) } are the same.
\end{remark}
\begin{example}\label{exa about Kelley's class sch}
It follows from (\ref{ax of infin})$\ar{ax}$ and (\ref{ax of extent})$\ar{ax}$
and Proposition \ref{Pro basic}\,(18) below that there are \math{a\kern0.37mm,\kern0.15mm b
\kern0.37mm,\kern0.15mm c} with \math{b\not=a} and \mathss36{a\kern0.37mm,\kern0.15mm b\in c}. For \mathss36{
A = \{\,x:\eexi y\,x=y\kern0.37mm\text{ and }\kern0.37mm y=a\,\}\yxbtext{15}1b }, we then
get from Remark \ref{rem about class sch}\,(t) and
Proposition \ref{Pro basic}\,(17) that for all \math{x\kern0.37mm,\kern0.15mm y} we have \vskip.4mm \centerline{$
x\in A\equivss22[ \KP{1.4} x\kern0.37mm$ set and $\kern0.37mm\eexi y\, x = y \kern0.37mm$ and $\kern0.37mm
y = a \KP{1.4} ]\equivss22 x = a \,$,} \inskipline{.2}0
and $\KP{13.7} y \in A \equivss22[ \KP{1.4} y \kern0.37mm$ set and $\kern0.37mm \eexi y\,
y = y\kern0.37mm $ and $\kern0.37mm y = a \KP{1.4} ]\equivss22 y\kern0.37mm$ set\kern0.37mm, \inskipline{.4}0 | 4,055 | 362,273 | en |
train | 0.47.9 | \end{axiom schema}
Above (\ref{ax of choice})$\ar{ax}$ is the {\sl global axiom of choice\kern0.15mm} and
(\ref{ax of infin})$\ar{ax}$ is the {\sl axiom of infinity\kern0.15mm}.
\begin{remark}\label{rem about class sch}
Among others, we accept as logical axioms the formulas \inskipline{.3}3
(1)$\ar{az}\,$ \ $\mfrk P\impss22\aall{\mfrk x}\,\mfrk P\,$, \KP{20}
(2)$\ar{az}\,$ \ $[ \KP{1.4} \aall{\mfrk x}\,\mfrk Q \KP{1.4} ]\impss22
\mfrk Q\,(\kern0.37mm\mfrk x\kern-0.3mm\lleftarrow\kern-0.3mm\mfrak F\kern0.37mm) \,$, \inskipline{.3}0
when \math{\mfrk x} is any variable symbol and \math{\mfrk F} is any term and
\math{\mfrk P\kern0.37mm,\mfrk Q} are any formulas such that for any variable symbol \math{
\mfrk y} having a free occurrence in \math{\mfrk F} the bound (\kern0.37mm i.e.\
non\kern0.37mm-\kern0.37mm free\kern0.15mm) occurrences of \math{\mfrk y} in \math{\mfrk Q} and \math{
\mfrk Q\,(\kern0.37mm\mfrk x\kern-0.3mm\leftarrow\kern-0.3mm\mfrak y\kern0.37mm)} are the same. Having these
logical axioms, we could give Axiom schema \ref{class axi} above a simpler
formulation than has the corresponding \cite[\erm{II}\kern0.37mm, p.\ 253]{Ky} which
in our notation would (\kern0.15mm as already a bit corrected\kern0.37mm) read as follows.
For any variable symbols \math{\mfrk x\kern0.37mm,\kern0.15mm\mfrk y\kern0.37mm,\kern0.15mm\mfrk z} and for
any formula \mathss30{\mfrk P} such that \math{\mfrk y} is the first one
distinct from \math{\mfrk x} and \mathss34{\mfrk z}, \,and not occurring free
in \mathss34{\mfrk P}, we accept as an axiom the formula \inskipline{.3}3
(t) \ $\aall{\mfrk z}\,\mfrk z\in\{\,\mfrk x:\mfrk P\,\}\yxbtext{15}1b\equivss22
\eexi{\mfrk y}\,\mfrk z\in\mfrk y\kern0.37mm$ and $\kern0.37mm
\mfrk P\,(\kern0.37mm\mfrk x\kern-0.3mm\leftarrow\kern-0.3mm\mfrak z\kern0.37mm)\,$. \inskipline{.3}0
However, this would make the system contradictory as shown in Example \ref{exa about Kelley's class sch}
below. One should put the additional restriction that the bound occurrences of \math{
\mfrk z} in $\kern0.37mm\mfrk P$ \linebreak
and \math{
\mfrk P\,(\kern0.37mm\mfrk x\kern-0.3mm\leftarrow\kern-0.3mm\mfrk z\kern0.37mm) } are the same.
\end{remark}
\begin{example}\label{exa about Kelley's class sch}
It follows from (\ref{ax of infin})$\ar{ax}$ and (\ref{ax of extent})$\ar{ax}$
and Proposition \ref{Pro basic}\,(18) below that there are \math{a\kern0.37mm,\kern0.15mm b
\kern0.37mm,\kern0.15mm c} with \math{b\not=a} and \mathss36{a\kern0.37mm,\kern0.15mm b\in c}. For \mathss36{
A = \{\,x:\eexi y\,x=y\kern0.37mm\text{ and }\kern0.37mm y=a\,\}\yxbtext{15}1b }, we then
get from Remark \ref{rem about class sch}\,(t) and
Proposition \ref{Pro basic}\,(17) that for all \math{x\kern0.37mm,\kern0.15mm y} we have \vskip.4mm \centerline{$
x\in A\equivss22[ \KP{1.4} x\kern0.37mm$ set and $\kern0.37mm\eexi y\, x = y \kern0.37mm$ and $\kern0.37mm
y = a \KP{1.4} ]\equivss22 x = a \,$,} \inskipline{.2}0
and $\KP{13.7} y \in A \equivss22[ \KP{1.4} y \kern0.37mm$ set and $\kern0.37mm \eexi y\,
y = y\kern0.37mm $ and $\kern0.37mm y = a \KP{1.4} ]\equivss22 y\kern0.37mm$ set\kern0.37mm, \inskipline{.4}0
whence taking \math{x=y=b\kern0.15mm}, we obtain \mathss38{[ \KPp1.4 b \kern0.37mm\text{ set }
\impss03 b\in A\impss33 b = a \KPp1.4 ] }, \,a {\sl contradiction\kern0.15mm}. The
formula \q{\math{\eexi y\, y = y\kern0.37mm\text{ and }\kern0.37mm y = a }} contains four
occurrences of \math{\Symboo yœ}. They are all bound and the second of them is
not present in \q{\math{\eexi y\, x = y\kern0.37mm$ and $\kern0.37mm y = a }}.
\end{example}
When we write a formula \mathss34{\mfrak P}, \,for example \q{\mathss00{ x =
\int_{\,\ssmb A}^{\KPt8\ssmb B}f\fvalss21 t\rmdss11 t}}, associated with the
writing appearance of \math{\mfrak P} we assume that there is an implicitly
understood well\kern0.15mm-\kern0.37mm order between the occurring variable symbols so that
e.g.\ it makes sense to refer to the first variable symbol occurring free in
the {\sl writing appearance\kern0.15mm} of \mathss34{\mfrak P}. This has nothing to do
with the intuitive \q{overall} well\kern0.15mm-\kern0.37mm order of all variable symbols of
our set theoretic language.
For example in the above formula the variable symbols \math{\Symboo xœ,
\Symboo\kern-0.63mm\smb A\kern-0.3mmœ,\Symboo\kern-0.2mm\smb B\kern-0.2mmœ,\Symboo\kern-0.63mm fœ} occur free, and \math{
\Symboo tœ} has two bound occurrences. We may assume that the order of the
free variable\kern0.37mm( symbol\kern0.37mm)\kern0.37mm s is precisely the one given above,
although it may not be perfectly clear which one of \math{
\Symboo\kern-0.63mm\smb A\kern-0.3mmœ} and \math{\Symboo\kern-0.2mm\smb B\kern-0.2mmœ} is before the other.
To avoid confusion, in such vague cases we refrain from referring to that
\q{implicit order}. In the above case we may then say that \math{\Symboo xœ}
is the first one, whereas in the case of the formula \q{\kern-0.63mm\mathss01{
\int_{\,\ssmb A}^{\KPt8\ssmb B}f\fvalss21 t\rmdss11 t = x }} we would not
speak of the free variable symbol that is in the first place in the writing
appearance.
Having the above preparative explanation, in order to have available a
convenient means of specifying functions, we give the following
\begin{def:al schema}\label{<T:F>}
Let \math{\mfrak T} be a term and \math{\mfrak F} a formula and \math{
\mfrak x\kern0.37mm,\afr x_1\kern0.15mm,\ldots\,\mfrak x\kern0.37mm\ai k\kern0.37mm,\kern0.15mm\afr y_1\kern0.15mm,\ldots\,
\mfrak y\kern0.37mm\ai l} distinct variable symbols such that \math{\afr x_1\kern0.15mm,
\ldots\,\mfrak x\kern0.37mm\ai k} are precisely the variable symbols which have a
free occurrence both in \math{\mfrak T} and \math{\mfrak F} and are distinct
from any of $\,\mfrak x\,,\kern0.15mm\afr y_1\kern0.15mm,$ $\ldots\,\mfrak y\kern0.37mm\ai l\kern0.37mm$.
Also assume that \math{\mfrak F} is of the form \math{
\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'074}\kern.2mm\,\mfrak p\kern0.15mm\,\mfrak x\kern0.37mm\kern.2mm\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'076}\kern0.15mm\,\mfrak E} or \math{
\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'074}\kern.2mm\,\mfrak k\kern0.15mm\,\mfrak p\kern0.15mm\,\mfrak x\kern0.37mm\kern.2mm\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'076}\kern0.15mm\,\mfrak E} where \math{
\mfrak p} is some predicate symbol and \math{\mfrak k} is a connective such
that in the writing appearance of $\kern0.37mm\mfrak F$ we have \math{\mfrak x} in
the first place. Then we let \vskip.2mm\centerline{$
\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:\afr y_1\kern0.15mm,\ldots\,\mfrak y\kern0.37mm\ai l\kern-0.63mm:
\mfrak F\kern0.15mm\,}=\{\,\mfrak z:\exi{\mfrak x\kern0.37mm,\afr x_1\kern0.15mm,\ldots\,
\mfrak x\kern0.37mm\ai k}\,\mfrak z=(\kern0.37mm\mfrak x\,,\mfrak T\kern0.37mm)\kern0.37mm$ and $\kern0.37mm
\mfrak F\,\}$} \vskip.2mm
\noindent where \math{\mfrak z} is the first variable symbol not occurring free in
\math{\mfrak T} or \math{\mfrak F\kern0.15mm}.
We also put \math{\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:\mfrak F\kern0.15mm\,}=\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:
\ :\mfrak F\kern0.15mm\,}} in the case where \math{\afr y_1\kern0.15mm,\ldots\,
\mfrak y\kern0.37mm\ai l} is an empty list, and further \math{\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:
\mfrak x\in\mfrak U\kern0.15mm\,}\subtext{old}=\{\,\mfrak z:\exi{\mfrak x}\,\mfrak z =
(\kern0.37mm\mfrak x\,,\mfrak T\kern0.37mm)\kern0.37mm$ and $\kern0.37mm\mfrak x\in\mfrak U\,\}} when \math{
\mfrak U} is any term not containing a free occurrence of \math{\mfrak x\kern0.15mm},
and \math{\mfrak z} is the first variable symbol distinct from \math{\mfrak x}
and not occurring free in \math{\mfrak T} or \math{\mfrak U\kern0.37mm}.
\end{def:al schema}
\begin{def:al schema}\label{uniqset}
We let \math{\uniqset\mfrak x:\mfrak P=\bigcap\,\{\,\mfrak z:\all{\mfrak x}\,
\mfrak P\kern0.15mm\Leftrightarrow\kern0.15mm\mfrak x=\mfrak z\,\}\kern0.15mm}, when \math{\mfrak x\kern0.37mm,
\mfrak z} are any distinct variable symbols and \math{\mfrak P} is any formula
where \math{\mfrak z} does not occur free. To get $\mfrak z$ uniquely chosen,
we may take as \math{\mfrak z} the first admissible w.r.t\ the intuitive
well\kern0.37mm-\kern0.37mm ordering of the variable symbols of our set theoretic language.
\end{def:al schema}
Under the agreement of unique choice of $\mfrak z$ above, for any formula $
\mfrak P$ and any distinct variable symbols $\mfrak x\kern0.37mm,\mfrak z$ with $
\mfrak z$ not occurring free in $\mfrak P\kern0.37mm$, now the formula $
\uniqset\mfrak x:\mfrak P=\bigcap\,\{\,\mfrak z:\all{\mfrak x}\,\mfrak P\kern0.15mm
\Leftrightarrow\kern0.15mm\mfrak x=\mfrak z\,\}\,$ is a theorem. \vskip.2mm | 3,776 | 362,273 | en |
train | 0.47.10 | \begin{def:al schema}\label{<T:F>}
Let \math{\mfrak T} be a term and \math{\mfrak F} a formula and \math{
\mfrak x\kern0.37mm,\afr x_1\kern0.15mm,\ldots\,\mfrak x\kern0.37mm\ai k\kern0.37mm,\kern0.15mm\afr y_1\kern0.15mm,\ldots\,
\mfrak y\kern0.37mm\ai l} distinct variable symbols such that \math{\afr x_1\kern0.15mm,
\ldots\,\mfrak x\kern0.37mm\ai k} are precisely the variable symbols which have a
free occurrence both in \math{\mfrak T} and \math{\mfrak F} and are distinct
from any of $\,\mfrak x\,,\kern0.15mm\afr y_1\kern0.15mm,$ $\ldots\,\mfrak y\kern0.37mm\ai l\kern0.37mm$.
Also assume that \math{\mfrak F} is of the form \math{
\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'074}\kern.2mm\,\mfrak p\kern0.15mm\,\mfrak x\kern0.37mm\kern.2mm\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'076}\kern0.15mm\,\mfrak E} or \math{
\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'074}\kern.2mm\,\mfrak k\kern0.15mm\,\mfrak p\kern0.15mm\,\mfrak x\kern0.37mm\kern.2mm\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'076}\kern0.15mm\,\mfrak E} where \math{
\mfrak p} is some predicate symbol and \math{\mfrak k} is a connective such
that in the writing appearance of $\kern0.37mm\mfrak F$ we have \math{\mfrak x} in
the first place. Then we let \vskip.2mm\centerline{$
\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:\afr y_1\kern0.15mm,\ldots\,\mfrak y\kern0.37mm\ai l\kern-0.63mm:
\mfrak F\kern0.15mm\,}=\{\,\mfrak z:\exi{\mfrak x\kern0.37mm,\afr x_1\kern0.15mm,\ldots\,
\mfrak x\kern0.37mm\ai k}\,\mfrak z=(\kern0.37mm\mfrak x\,,\mfrak T\kern0.37mm)\kern0.37mm$ and $\kern0.37mm
\mfrak F\,\}$} \vskip.2mm
\noindent where \math{\mfrak z} is the first variable symbol not occurring free in
\math{\mfrak T} or \math{\mfrak F\kern0.15mm}.
We also put \math{\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:\mfrak F\kern0.15mm\,}=\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:
\ :\mfrak F\kern0.15mm\,}} in the case where \math{\afr y_1\kern0.15mm,\ldots\,
\mfrak y\kern0.37mm\ai l} is an empty list, and further \math{\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:
\mfrak x\in\mfrak U\kern0.15mm\,}\subtext{old}=\{\,\mfrak z:\exi{\mfrak x}\,\mfrak z =
(\kern0.37mm\mfrak x\,,\mfrak T\kern0.37mm)\kern0.37mm$ and $\kern0.37mm\mfrak x\in\mfrak U\,\}} when \math{
\mfrak U} is any term not containing a free occurrence of \math{\mfrak x\kern0.15mm},
and \math{\mfrak z} is the first variable symbol distinct from \math{\mfrak x}
and not occurring free in \math{\mfrak T} or \math{\mfrak U\kern0.37mm}.
\end{def:al schema}
\begin{def:al schema}\label{uniqset}
We let \math{\uniqset\mfrak x:\mfrak P=\bigcap\,\{\,\mfrak z:\all{\mfrak x}\,
\mfrak P\kern0.15mm\Leftrightarrow\kern0.15mm\mfrak x=\mfrak z\,\}\kern0.15mm}, when \math{\mfrak x\kern0.37mm,
\mfrak z} are any distinct variable symbols and \math{\mfrak P} is any formula
where \math{\mfrak z} does not occur free. To get $\mfrak z$ uniquely chosen,
we may take as \math{\mfrak z} the first admissible w.r.t\ the intuitive
well\kern0.37mm-\kern0.37mm ordering of the variable symbols of our set theoretic language.
\end{def:al schema}
Under the agreement of unique choice of $\mfrak z$ above, for any formula $
\mfrak P$ and any distinct variable symbols $\mfrak x\kern0.37mm,\mfrak z$ with $
\mfrak z$ not occurring free in $\mfrak P\kern0.37mm$, now the formula $
\uniqset\mfrak x:\mfrak P=\bigcap\,\{\,\mfrak z:\all{\mfrak x}\,\mfrak P\kern0.15mm
\Leftrightarrow\kern0.15mm\mfrak x=\mfrak z\,\}\,$ is a theorem. \vskip.2mm
One quickly deduces that if a unique {\it set\kern0.37mm} \math{\mfrak x} exists with
\math{\mfrak P}, then \math{\uniqset\mfrak x:\mfrak P=\mfrak x\kern0.15mm}. In all
other cases, i.e.\ when there is no \math{\mfrak x\in\hbox{\font\SweD =cmssbx10\SweD U}{}} with \math{
\mfrak P}, or if (with $\mfrak y$ being a variable symbol not occurring in \math{
\mfrak P}) there are \math{\mfrak x\kern0.37mm,\mfrak y\in\hbox{\font\SweD =cmssbx10\SweD U}{}} with \math{
\mfrak x\not=\mfrak y} and \math{\mfrak P} and \math{
\mfrak P\,(\kern0.37mm\mfrak x\kern-0.3mm\leftarrow\kern-0.3mm\mfrak y\kern0.37mm)\kern0.15mm}, substitution in
places of free occurrence, then \math{ \label{END set th extract}
\uniqset\mfrak x:\mfrak P = \bigcap\kern0.37mm\emptyset = \hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm}. \vskip.4mm
Below in Definitions \ref{misc defs}\,(9) we have \math{\raise1.35mm\hbox{\font\SweD =cmr5\SweD c}\kern-.15mm\infty =
(\kern0.37mm 0\ar 1\kern0.15mm,\lower1.05mm\hbox{$^+$}\infty\ar 1\kern0.37mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1) } the
{\sl complex infinity\kern0.15mm} for some \math{\lower1.05mm\hbox{$^+$}\infty\ar 1\in\ovbbR\ar 1} whose
exact construction we here omit. Also omitting the precise definition, note
that \math{|\,\zeta\,|\suba} is the standard \erm Euclidean absolute value of
any quaternion \math{\zeta} and that we below usually have \math{\zeta} a real
or complex number.
\begin{definitions}\label{misc defs}
(1) \ $\bbI=[\KPp1.1 0\,,\kern0.07mm 1\KPt9] \KP1 $, \KP{
8.3}
(2) \ $\mathbb J=\openIval{\KPt5 0\,,\kern0.07mm 1\kern0.15mm} \KP1 $, \inskipline{.5}2
(3) \ $\rbb R^+=\mathbb R\capss41\{\,t:0<t\KPt9\} \KP1 $, \KP{7.5}
(4) \ $\lbb R_+=\mathbb R\capss41\{\,t:0\le t\KPt9\} \KP1 $, \inskipline{.5}2
(5) \ $
\mathbb Z=
\bigcap\KPt8\{\,N\kern-0.3mm:
0\in N\subseteq\mathbb R\text{ and }\kern0.37mm
\aall{n\in N}\,
\{\,n-1\kern0.37mm,\kern0.15mm n+1\,\}\subseteq N\KP1\}
\KP1 $, \inskipline{.5}2
(6) \ $\rbb Z^+=\mathbb Z\capss41\{\,n:0<n\KPt9\} \KP1 $, \KP{6.5}
(7) \ $\lbb Z_+=\mathbb Z\capss41\{\,n:0\le n\KPt9\} \KP1 $, \inskipline{.5}2
(8) \ $p\,^* = \uniqset t:[\KPp1.4 1 < p < \lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm t =
(\kern0.37mm 1 - p\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm\big){}^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\KP1\big]\kern0.37mm$ or \inskipline0{31.6}
$[\KPp1.4 p=1\kern0.37mm$ and $\kern0.37mm t=\lower1.05mm\hbox{$^+$}\infty\KPp1.4]\kern0.37mm$ or $\kern0.37mm
[\KPp1.4 p=\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm t=1\KPp1.4] \KP1 $, \inskipline{.5}2
(9) \ $\Abrs33^p = \uniqset\chi:p\in\rbb R^+\kern0.15mm$ and \inskipline{.2}{28.65}
$\chi \kern0.37mm = \kern0.37mm \{\kern0.15mm\kern.2mm\lower1.05mm\hbox{$^-$}infty\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm,\raise1.35mm\hbox{\font\SweD =cmr5\SweD c}\kern-.15mm\infty\,\}\kern-.2mm\times\kern-.2mm
\{\kern0.15mm\lower1.05mm\hbox{$^+$}\infty\,\} \cupss21 \big\langle\KP{1.2}
|\,\zeta\,|\suba\RHB{.25}{^p} \kern-0.3mm : \zeta\in\mathbb H\KPp1.2 \rangle \KP1 $, \inskipline{.5}2
(10) \ $\scrb8 T\,$ is a {\it topology\kern0.37mm}
$\equivss33 \emptyset\not=\scrb8 T\in\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm$ and \inskipline{.2}{10.5}
$\aall{\Cal A}\,\Cal A\subseteq\scrb8 T\impss33
\bigcup\,\Cal A\in\scrb8 T\kern0.37mm\text{ and }\kern0.37mm
[\KP{1.2} \Cal A\not=\emptyset\kern0.37mm\text{ and }\kern0.37mm
\Cal A\kern0.37mm\text{ is finite }\Rightarrow\kern0.37mm\bigcap\,\Cal A\in
\scrb8 T \KP{1.5} ] \KP1$, \inskipline{.5}2
(11) \ $\scrb8 T\,$ is a {\it separable\kern0.37mm} topology $\equivss33
\scrb8 T\,$ is a topology and \inskipline{.2}{10.5}
$\eexi D\,D\kern0.37mm$ is countable and
$\kern0.37mm\aall U\,U\in\scrb8 T\impss33 D\capss33 U\not=\emptyset\kern0.37mm\text{ or }\kern0.37mm
U=\emptyset\KPt8$, \inskipline{.5}2
(12) \ $\scrb8 T\,$ is a {\it compact\kern0.37mm} topology $\equivss33
\scrb8 T\,$ is a topology and \inskipline{.2}{10.5}
$\aall{\Cal A}\,\eexi{\Cal B}\,
\Cal A\subseteq\scrb8 T\impss33
\Cal B\subseteq\Cal A\kern0.37mm$ and $\kern0.37mm\Cal B\kern0.37mm$ is finite and \inskipline{.2}{10.5}
$\kern0.37mm[\KP{1.6}
\bigcup\,\Cal A\subseteq\bigcup\,\Cal B\kern0.37mm$ or $\kern0.37mm
\bigcup\,\Cal A\not=\bigcup\,\scrb8 T \KP{1.5} ] \KP1$, \inskipline{.5}2
(13) \ $\scrmt A\kern0.37mm$ is {\it disjoint\kern0.37mm} $\equivss33\aall{A\,,\kern0.15mm B}\,
A\,,\kern0.15mm B\in\scrmt A\impss33 A=B\kern0.37mm$ or $\kern0.37mm A\capss32 B=\emptyset \,$, \inskipline{.5}2
(14) \ $\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A=\{\,B:B\subseteq A\KP1\} \KP1 $, \KP{7.2}
(15) \ $\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 B=\{\,A\capss32 B : A\in\scrmt A\KP1\} \KP1 $, \inskipline{.5}2 | 4,017 | 362,273 | en |
train | 0.47.11 | (8) \ $p\,^* = \uniqset t:[\KPp1.4 1 < p < \lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm t =
(\kern0.37mm 1 - p\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm\big){}^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\KP1\big]\kern0.37mm$ or \inskipline0{31.6}
$[\KPp1.4 p=1\kern0.37mm$ and $\kern0.37mm t=\lower1.05mm\hbox{$^+$}\infty\KPp1.4]\kern0.37mm$ or $\kern0.37mm
[\KPp1.4 p=\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm t=1\KPp1.4] \KP1 $, \inskipline{.5}2
(9) \ $\Abrs33^p = \uniqset\chi:p\in\rbb R^+\kern0.15mm$ and \inskipline{.2}{28.65}
$\chi \kern0.37mm = \kern0.37mm \{\kern0.15mm\kern.2mm\lower1.05mm\hbox{$^-$}infty\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm,\raise1.35mm\hbox{\font\SweD =cmr5\SweD c}\kern-.15mm\infty\,\}\kern-.2mm\times\kern-.2mm
\{\kern0.15mm\lower1.05mm\hbox{$^+$}\infty\,\} \cupss21 \big\langle\KP{1.2}
|\,\zeta\,|\suba\RHB{.25}{^p} \kern-0.3mm : \zeta\in\mathbb H\KPp1.2 \rangle \KP1 $, \inskipline{.5}2
(10) \ $\scrb8 T\,$ is a {\it topology\kern0.37mm}
$\equivss33 \emptyset\not=\scrb8 T\in\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm$ and \inskipline{.2}{10.5}
$\aall{\Cal A}\,\Cal A\subseteq\scrb8 T\impss33
\bigcup\,\Cal A\in\scrb8 T\kern0.37mm\text{ and }\kern0.37mm
[\KP{1.2} \Cal A\not=\emptyset\kern0.37mm\text{ and }\kern0.37mm
\Cal A\kern0.37mm\text{ is finite }\Rightarrow\kern0.37mm\bigcap\,\Cal A\in
\scrb8 T \KP{1.5} ] \KP1$, \inskipline{.5}2
(11) \ $\scrb8 T\,$ is a {\it separable\kern0.37mm} topology $\equivss33
\scrb8 T\,$ is a topology and \inskipline{.2}{10.5}
$\eexi D\,D\kern0.37mm$ is countable and
$\kern0.37mm\aall U\,U\in\scrb8 T\impss33 D\capss33 U\not=\emptyset\kern0.37mm\text{ or }\kern0.37mm
U=\emptyset\KPt8$, \inskipline{.5}2
(12) \ $\scrb8 T\,$ is a {\it compact\kern0.37mm} topology $\equivss33
\scrb8 T\,$ is a topology and \inskipline{.2}{10.5}
$\aall{\Cal A}\,\eexi{\Cal B}\,
\Cal A\subseteq\scrb8 T\impss33
\Cal B\subseteq\Cal A\kern0.37mm$ and $\kern0.37mm\Cal B\kern0.37mm$ is finite and \inskipline{.2}{10.5}
$\kern0.37mm[\KP{1.6}
\bigcup\,\Cal A\subseteq\bigcup\,\Cal B\kern0.37mm$ or $\kern0.37mm
\bigcup\,\Cal A\not=\bigcup\,\scrb8 T \KP{1.5} ] \KP1$, \inskipline{.5}2
(13) \ $\scrmt A\kern0.37mm$ is {\it disjoint\kern0.37mm} $\equivss33\aall{A\,,\kern0.15mm B}\,
A\,,\kern0.15mm B\in\scrmt A\impss33 A=B\kern0.37mm$ or $\kern0.37mm A\capss32 B=\emptyset \,$, \inskipline{.5}2
(14) \ $\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A=\{\,B:B\subseteq A\KP1\} \KP1 $, \KP{7.2}
(15) \ $\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 B=\{\,A\capss32 B : A\in\scrmt A\KP1\} \KP1 $, \inskipline{.5}2
(16) \ $\nsTbb_R=\big\{\kern0.37mm\bigcup\,\scrmt A:\scrmt A\subseteq\big\{\,
\openIval{\kern0.37mm s\kern0.37mm,\kern0.07mm t\kern0.37mm}:\kern.2mm\lower1.05mm\hbox{$^-$}infty < s < t < \lower1.05mm\hbox{$^+$}\infty\KPt9\}
\kern0.15mm\} \KP1 $, \inskipline{.5}2
(17) \ $\barscTbb_R=
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm[\kern0.37mm\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\lower1.05mm\hbox{$^+$}\infty\KP1]\capss51\{\KPt8 U\kern-0.3mm:
U\cap\ssbb40 R\in\nsTbb_R\kern0.37mm$ and $\kern0.37mm\eexi{s\kern0.37mm,\kern0.15mm r\in\mathbb R}\,$
\inskipline{.2}{21}
$
[\KPp1.4 \lower1.05mm\hbox{$^+$}\infty\in U\impss33
{\kern0.37mm]}\KP1 s\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\KPt9]
\subseteq U\KPp1.4]\kern0.37mm$ and $\kern0.37mm[\KPp1.4 \kern.2mm\lower1.05mm\hbox{$^-$}infty\in U\impss33
[\kern0.37mm\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\kern0.07mm r\KP1{[\kern0.37mm}
\subseteq U\KPp1.4]\KP1\big\} \KP1 $, \inskipline{.5}2
(18) \ $f\fvalss20 x=
\bigcap\KPt8\{\,y:\aall z\,
(\kern0.37mm x\kern0.37mm,\kern0.07mm z\kern0.37mm)\in f\equivss33 y=z\,\} \KP1 $, \inskipline{.5}2
(19) \ $f\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A=\{\,y:
\eexi{x\in A}\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in f\KP1\} \KP1 $, \KP4
(20) \ $f\KPt8[\KPt8 A\KPt9]=f\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A \KPt8 $, \inskipline{.5}2
(21) \ $f\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern-0.3mm\scrmt A=
\{\,f\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A:A\in\scrmt A\KP1\} \KP1 $, \KP{18.2}
(22) \ $f\invss40=\{\,(\kern0.37mm y\kern0.37mm,\kern0.07mm x\kern0.37mm):
(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in f\KP1\} \KP1 $, \inskipline{.5}2
(23) \ ${{}^{}{\rm dom}\,{}_{{}^{}}} f=\{\,x:\eexi y\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in f\KP1\} \KP1 $, \KP8
(24) \ ${}^{}{\rm rng}\,{}_{{}^{}} f=\{\,y:\eexi x\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in f\KP1\} \KP1 $, \inskipline{.5}2
(25) \ $E\Reit3=\uniqset F:\eexi{a\kern0.37mm,\kern0.15mm c\,,\kern0.15mm\scrmt S}\, E =
(\kern0.37mm a\kern0.37mm,\kern0.15mm c\,,\kern0.07mm\scrmt S\kern0.37mm)\kern0.37mm$ and \mathss38{F =
(\kern0.37mm a\kern0.37mm,\kern0.15mm c\KPp1.1|\KP1(\ssbb40 R\times\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm)\,,\kern0.07mm\scrmt S\kern0.37mm) }, \inskipline{.5}2
(26) \ $^A\,B = A\times B\capss31\{\,f:f\kern0.37mm\text{ is a function and }\kern0.37mm
A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} f\KP1\} \KP1 $, \inskipline{.5}2
(27) \ $\prod{_{_{\kern-.3mm\bold c\kern.15mm}}}\kern0.15mm\bmii8 A \kern0.15mm = \kern0.15mm ^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii6 A}\,\hbox{\font\SweD =cmssbx10\SweD U}{}\capss31\{\,
x:\aall{i\kern0.37mm,\kern0.15mm\xi}\,(\kern0.37mm i\kern0.37mm,\kern0.07mm\xi\kern0.37mm)\in x\impss33\xi\in
\bmii8 A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\KPt9\} \KP1 $, \inskipline{.5}2
(28) \ $\bosy x\to x\kern0.37mm$ in top \math{\scrmt T\equivss33\bosy x \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\bigcup\,\scrmt T } and \math{x\in\bigcup\,\scrmt T } and \inskipline0{24.2}
$\aall{\kern0.37mm U}\,\eexi{\smb N}\,x\in\kern0.15mm U\kern-0.3mm\in\scrmt T \impss33 \smb N \in
\mathbb No \kern0.37mm$ and \mathss30{\bosy x\KP1[\KPp1.1\mathbb No\kern-0.3mm\setminus\smb N\KP1]
\subseteq \kern0.15mm U}, \inskipline{.5}2
(29) \ $E\subsigrs04=\uniqset F:\eexi{\bosy K}\,\bosy K\kern0.37mm$ is a
topological division ring and \math{\Bnull_{\bosy K} = 0 } and \inskipline0{14}
$E\in\tvsps0(K) \kern0.37mm $ and \math{\aall{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm,\kern0.15mm I\kern0.15mm,\kern0.15mm\scrmt T}\,
I = \Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } and \inskipline0{31.5} | 3,771 | 362,273 | en |
train | 0.47.12 | (21) \ $f\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern-0.3mm\scrmt A=
\{\,f\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A:A\in\scrmt A\KP1\} \KP1 $, \KP{18.2}
(22) \ $f\invss40=\{\,(\kern0.37mm y\kern0.37mm,\kern0.07mm x\kern0.37mm):
(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in f\KP1\} \KP1 $, \inskipline{.5}2
(23) \ ${{}^{}{\rm dom}\,{}_{{}^{}}} f=\{\,x:\eexi y\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in f\KP1\} \KP1 $, \KP8
(24) \ ${}^{}{\rm rng}\,{}_{{}^{}} f=\{\,y:\eexi x\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in f\KP1\} \KP1 $, \inskipline{.5}2
(25) \ $E\Reit3=\uniqset F:\eexi{a\kern0.37mm,\kern0.15mm c\,,\kern0.15mm\scrmt S}\, E =
(\kern0.37mm a\kern0.37mm,\kern0.15mm c\,,\kern0.07mm\scrmt S\kern0.37mm)\kern0.37mm$ and \mathss38{F =
(\kern0.37mm a\kern0.37mm,\kern0.15mm c\KPp1.1|\KP1(\ssbb40 R\times\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm)\,,\kern0.07mm\scrmt S\kern0.37mm) }, \inskipline{.5}2
(26) \ $^A\,B = A\times B\capss31\{\,f:f\kern0.37mm\text{ is a function and }\kern0.37mm
A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} f\KP1\} \KP1 $, \inskipline{.5}2
(27) \ $\prod{_{_{\kern-.3mm\bold c\kern.15mm}}}\kern0.15mm\bmii8 A \kern0.15mm = \kern0.15mm ^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii6 A}\,\hbox{\font\SweD =cmssbx10\SweD U}{}\capss31\{\,
x:\aall{i\kern0.37mm,\kern0.15mm\xi}\,(\kern0.37mm i\kern0.37mm,\kern0.07mm\xi\kern0.37mm)\in x\impss33\xi\in
\bmii8 A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\KPt9\} \KP1 $, \inskipline{.5}2
(28) \ $\bosy x\to x\kern0.37mm$ in top \math{\scrmt T\equivss33\bosy x \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\bigcup\,\scrmt T } and \math{x\in\bigcup\,\scrmt T } and \inskipline0{24.2}
$\aall{\kern0.37mm U}\,\eexi{\smb N}\,x\in\kern0.15mm U\kern-0.3mm\in\scrmt T \impss33 \smb N \in
\mathbb No \kern0.37mm$ and \mathss30{\bosy x\KP1[\KPp1.1\mathbb No\kern-0.3mm\setminus\smb N\KP1]
\subseteq \kern0.15mm U}, \inskipline{.5}2
(29) \ $E\subsigrs04=\uniqset F:\eexi{\bosy K}\,\bosy K\kern0.37mm$ is a
topological division ring and \math{\Bnull_{\bosy K} = 0 } and \inskipline0{14}
$E\in\tvsps0(K) \kern0.37mm $ and \math{\aall{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm,\kern0.15mm I\kern0.15mm,\kern0.15mm\scrmt T}\,
I = \Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } and \inskipline0{31.5}
$\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm=\seqss03{\kern-0.3mm\seqss33{u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x:u\in I}:x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} \kern0.37mm$ and \mathss39{
\scrmt T=\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.15mm(\kern0.37mm
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\expnota^\kern0.15mm I\kern0.15mm]_{ti}\big) } \inskipline0{90.5}
$\impss03 F = (\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm,\kern0.07mm\scrmt T\,) \KP1 $.
\end{definitions}
About the {\sl weakening\kern0.15mm} \math{E\subsigrs04} of \math{E} in
Definitions \ref{misc defs}\,(29) above we note the following. If \math{E} is
a topological vector space over a topological division ring \mathss32{\bosy K
}, \,there may exist another topological division ring \math{\bosy K\kern-0.3mm\ar 1}
with \mathss38{E\in\tvsps9(K\kern-0.3mm\ar 1\kern-0.3mm) }. In every case then \math{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K={{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K\aR 1} holds, but \math{\bosy K} and \math{
\bosy K\kern-0.3mm\ar 1} may possess different zero elements if \math{ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E =
\{\,\Bnull_E\} } holds. Then the condition \math{\Bnull_{\bosy K}=0=
\Bnull_{\kern0.15mm\aars{\bosy K\kern-0.2mm}_1} } excludes this possibility. If \math{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E=\{\,\Bnull_E\} } holds, for \math{ I =
\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } and \mathss30{ n =
\{\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} } then necessarily \math{ I =
\{\kern0.37mm n\kern0.37mm\} } holds, and we get \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \{\KPt8(\,\Bnull_E\kern0.37mm,\kern0.15mm
\{\,(\kern0.37mm n\kern0.37mm,\kern0.07mm 0\kern0.37mm)\,\}\kern0.37mm\sbig)2\kern0.37mm\big\} } and further \mathss06{
\scrmt T=\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E }. Hence in this case \math{\scrmt T} is uniquely
determined although \mathss30{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\not=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\kern-0.3mm\ar 1 } may
hold. If \math{I\not=\{\kern0.37mm n\kern0.37mm\} } holds, then one deduces from the
postulates in the definition of a topological vector space that we necessarily
have \math{\bosy K=\bosy K\kern-0.3mm\ar 1 } and consequently again \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm,\kern0.07mm\scrmt T\,) } is uniquely determined.
Thus the above definition of \math{E\subsigrs04} is meaningful for precisely
those topological vector spaces \math{E} that are \q{over} some topological
division ring whose zero element is the same as that of the quaternionic one.
For more general cases one has to use a more complicated notation e.g.\ from \math{
E\subsigrs04\kern0.37mm\langle\,\bosy K=E\subsigrs04\kern0.37mm(\kern0.15mm I\kern0.37mm) } for \math{I} as
above, once the appropriate additional definition is specified.
In \ref{misc defs}\,(18) above \math{f\fvalss20 x} is the function value of \math{
f} at \math{x} which usually is written in a more complicated manner
\q{$f\kern0.37mm(x)$}, and possibly having a different formal definition as for
example in \cite[Definition 68\kern0.37mm, p.\ 261]{Ky}\,. We further state some
basic definitions and their simple consequences without proofs in the
following
\begin{proposition}\label{Pro basic}
$\null$ {\rm \inskipline{.5}2
(1) \ }$\emptyset=\{\,x:x\not=x\,\} \KP1 ${\rm, \KP{24}
(2) \ }$\hbox{\font\SweD =cmssbx10\SweD U}{}=\{\,x:x=x\,\} \KP1 ${\rm, \inskipline{.5}2
(3) \ $ \roman{pr}\ar 1 =
\{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm x\kern0.37mm):x\kern0.37mm,\kern0.15mm y\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \KP1 $, \KP5
(4) \ }$\roman{pr}\ar 2 =
\{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm y\kern0.37mm):x\kern0.37mm,\kern0.15mm y\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \KP1
${\rm, \inskipline{.5}2
(5) \ }$\roman{ev}=\{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm u\kern0.37mm,\kern0.07mm y\kern0.37mm):u\kern0.37mm\text{ is a
function and }\kern0.37mm(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in u\,\} \KP1 ${\rm, \inskipline{.5}2
(6) \ }$\roman{ev}\sbi{\kern0.15mm\emath x} =
\seqss33{u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x:u\kern0.37mm\text{ is a function}}$ \inskipline{.2}{14.45} | 3,964 | 362,273 | en |
train | 0.47.13 | Thus the above definition of \math{E\subsigrs04} is meaningful for precisely
those topological vector spaces \math{E} that are \q{over} some topological
division ring whose zero element is the same as that of the quaternionic one.
For more general cases one has to use a more complicated notation e.g.\ from \math{
E\subsigrs04\kern0.37mm\langle\,\bosy K=E\subsigrs04\kern0.37mm(\kern0.15mm I\kern0.37mm) } for \math{I} as
above, once the appropriate additional definition is specified.
In \ref{misc defs}\,(18) above \math{f\fvalss20 x} is the function value of \math{
f} at \math{x} which usually is written in a more complicated manner
\q{$f\kern0.37mm(x)$}, and possibly having a different formal definition as for
example in \cite[Definition 68\kern0.37mm, p.\ 261]{Ky}\,. We further state some
basic definitions and their simple consequences without proofs in the
following
\begin{proposition}\label{Pro basic}
$\null$ {\rm \inskipline{.5}2
(1) \ }$\emptyset=\{\,x:x\not=x\,\} \KP1 ${\rm, \KP{24}
(2) \ }$\hbox{\font\SweD =cmssbx10\SweD U}{}=\{\,x:x=x\,\} \KP1 ${\rm, \inskipline{.5}2
(3) \ $ \roman{pr}\ar 1 =
\{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm x\kern0.37mm):x\kern0.37mm,\kern0.15mm y\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \KP1 $, \KP5
(4) \ }$\roman{pr}\ar 2 =
\{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm y\kern0.37mm):x\kern0.37mm,\kern0.15mm y\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \KP1
${\rm, \inskipline{.5}2
(5) \ }$\roman{ev}=\{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm u\kern0.37mm,\kern0.07mm y\kern0.37mm):u\kern0.37mm\text{ is a
function and }\kern0.37mm(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in u\,\} \KP1 ${\rm, \inskipline{.5}2
(6) \ }$\roman{ev}\sbi{\kern0.15mm\emath x} =
\seqss33{u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x:u\kern0.37mm\text{ is a function}}$ \inskipline{.2}{14.45}
${} = \{\,(\kern0.37mm u\,,\kern0.07mm y\kern0.37mm):u\kern0.37mm\text{ is a function and }\kern0.37mm
(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in u\KPt8\} \KP1 ${\rm, \inskipline{.5}2
(7) \ }$\scrmt A\,,\kern0.15mm\scrmt B$ disjoint iff $\,\scrmt A$ and $\,\scrmt B$
disjoint iff $\,\scrmt A$ and $\,\scrmt B$ are disjoint iff \inskipline0{52.2}
$\scrmt A$ is disjoint and $\,\scrmt B$ is disjoint{\kern0.15mm\rm, \inskipline{.5}2
(8) \ }$ \newcommand\opair[2]{\hbox{\kern.#1mm\kern-.2mm\font\≈=cmtt10\≈,\kern-.2mm\kern.#2mm}}
x \opair14 y = (\kern0.15mm x\kern0.37mm,\kern0.07mm y\kern0.37mm) =
\{\kern0.15mm\{\,x\kern0.37mm,\kern0.15mm y\,\}\,,\kern0.15mm\{\kern0.37mm y\kern0.37mm\}\kern0.15mm\} \KP1 ${\rm, \KP{3.9}
(9) \ }$(\kern0.15mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm) =
((\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\kern0.37mm,\kern0.07mm z\kern0.37mm) \KP1 ${\rm, \inskipline{.5}2
(10) \ }$(\kern0.37mm x\,;\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm) =
(\kern0.37mm x\kern0.37mm,\kern0.07mm(\kern0.37mm y\kern0.37mm,\kern0.07mm z\kern0.37mm)) \KP1 ${\rm, \KP{15}
(11) \ }$(\kern0.15mm x\kern0.37mm,\kern0.07mm y\,;\kern0.07mm u\kern0.37mm,\kern0.07mm v\kern0.37mm) =
(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm(\kern0.37mm u\kern0.37mm,\kern0.07mm v\kern0.37mm)) \KP1 ${\rm, \inskipline{.5}2
(12) \ }$\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z = \bigcup\bigcup\kern0.15mm z\setminus\bigcup\bigcap\kern0.15mm z
\kern0.15mm\cup\kern0.15mm \bigcap\bigcup\kern0.15mm z ${\,\rm, \KP{3.8}
(13) \ }$\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z=\bigcap\bigcap\kern0.15mm z ${\,\rm, \inskipline{.5}2
(14) \ }$\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm z = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z\kern0.37mm) ${\KP1\rm, \KP{22.8}
(15) \ }$\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z = \tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z\kern0.37mm) ${\KP1\rm, \inskipline{.5}2
(16) \ }$z = (\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in\hbox{\font\SweD =cmssbx10\SweD U}{}\impss33 x = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z$ and $\kern0.37mm
y = \tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z ${\,\rm, \inskipline{.5}2
(17) \ }$x\kern0.37mm$ is a set $\equivss22 x\kern0.37mm$ a set $\equivss22
x\kern0.37mm$ set $\equivss22 \eexi y\,x\in y ${\kern0.37mm\rm, \inskipline{.5}2
(18) \ }$x \not= y\equivss22$ not $[\KP{1.4} x=y \KP{1.4} ] \KP1 $.
\end{proposition}
Observe for example that if \math{E=(\kern0.37mm a\kern0.37mm,\kern0.15mm c\,,\kern0.07mm\scrmt S\kern0.37mm)\not=
\hbox{\font\SweD =cmssbx10\SweD U}{} } with \math{c} a function \mathss34{R\times S\to S}, \,then \
${{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E={{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.37mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm)))=
{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.37mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.37mm a\kern0.37mm,\kern0.15mm c\kern0.37mm)))$ \inskipline0{28}
${}={{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}} c\kern0.37mm)={{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm R\times S\kern0.37mm)=R\kern0.37mm$ {\sl if\kern0.15mm} \math{
S \not= \emptyset } holds. \vskip.5mm
To see that the above given convention of \q{\mathss00{u\le v}} having a
meaning for both extended real numbers and extended real number valued
functions \math{u\kern0.37mm,\kern0.15mm v} does not create any contradiction in our logical
system, we need the following
\begin{lemma}
For every function $\,u$ with $\,{}^{}{\rm rng}\,{}_{{}^{}} u\subseteq\ovbbR$ it holds that $\,u\not\in
\ovbbR \, $.
\end{lemma}
\begin{proof} The {\sl regularity axiom\kern0.15mm} (\ref{ax of regularity})$\ar{ax}$
on page \pageref{ax of regularity} above, cf.\ \cite[\erm{VII}\kern0.15mm, p.\ 266]{Ky}
or \cite[\erm{ZF\,}9\kern0.37mm, p.\ 401]{Du}\,, has the simple consequence that
there {\sl do not exist\kern0.15mm} any \mathss30{x\ar 0\,,\kern0.15mm x\ar 1\kern0.15mm,\kern0.15mm x\ar 2}
such that \math{x\ar 0\in x\ar 1\in x\ar 2\in x\ar 0} holds. We show that this
will be contradicted if there exists a function \math{u} with \math{{}^{}{\rm rng}\,{}_{{}^{}} u\subseteq
\ovbbR} and \mathss37{u\in\ovbbR}. Indeed, then there is \math{r} with \inskipline{.42}{17.72}
$u = (\kern0.37mm r\kern0.15mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.07mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)
= (\kern0.37mm r\kern0.15mm,\kern0.07mm 0\ar 1\kern0.15mm,\kern0.15mm(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1))$ \inskipline{.2}{19.8} | 3,859 | 362,273 | en |
train | 0.47.14 | (16) \ }$z = (\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in\hbox{\font\SweD =cmssbx10\SweD U}{}\impss33 x = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z$ and $\kern0.37mm
y = \tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z ${\,\rm, \inskipline{.5}2
(17) \ }$x\kern0.37mm$ is a set $\equivss22 x\kern0.37mm$ a set $\equivss22
x\kern0.37mm$ set $\equivss22 \eexi y\,x\in y ${\kern0.37mm\rm, \inskipline{.5}2
(18) \ }$x \not= y\equivss22$ not $[\KP{1.4} x=y \KP{1.4} ] \KP1 $.
\end{proposition}
Observe for example that if \math{E=(\kern0.37mm a\kern0.37mm,\kern0.15mm c\,,\kern0.07mm\scrmt S\kern0.37mm)\not=
\hbox{\font\SweD =cmssbx10\SweD U}{} } with \math{c} a function \mathss34{R\times S\to S}, \,then \
${{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E={{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.37mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm)))=
{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.37mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.37mm a\kern0.37mm,\kern0.15mm c\kern0.37mm)))$ \inskipline0{28}
${}={{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}} c\kern0.37mm)={{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm R\times S\kern0.37mm)=R\kern0.37mm$ {\sl if\kern0.15mm} \math{
S \not= \emptyset } holds. \vskip.5mm
To see that the above given convention of \q{\mathss00{u\le v}} having a
meaning for both extended real numbers and extended real number valued
functions \math{u\kern0.37mm,\kern0.15mm v} does not create any contradiction in our logical
system, we need the following
\begin{lemma}
For every function $\,u$ with $\,{}^{}{\rm rng}\,{}_{{}^{}} u\subseteq\ovbbR$ it holds that $\,u\not\in
\ovbbR \, $.
\end{lemma}
\begin{proof} The {\sl regularity axiom\kern0.15mm} (\ref{ax of regularity})$\ar{ax}$
on page \pageref{ax of regularity} above, cf.\ \cite[\erm{VII}\kern0.15mm, p.\ 266]{Ky}
or \cite[\erm{ZF\,}9\kern0.37mm, p.\ 401]{Du}\,, has the simple consequence that
there {\sl do not exist\kern0.15mm} any \mathss30{x\ar 0\,,\kern0.15mm x\ar 1\kern0.15mm,\kern0.15mm x\ar 2}
such that \math{x\ar 0\in x\ar 1\in x\ar 2\in x\ar 0} holds. We show that this
will be contradicted if there exists a function \math{u} with \math{{}^{}{\rm rng}\,{}_{{}^{}} u\subseteq
\ovbbR} and \mathss37{u\in\ovbbR}. Indeed, then there is \math{r} with \inskipline{.42}{17.72}
$u = (\kern0.37mm r\kern0.15mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.07mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)
= (\kern0.37mm r\kern0.15mm,\kern0.07mm 0\ar 1\kern0.15mm,\kern0.15mm(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1))$ \inskipline{.2}{19.8}
${}= ((\kern0.37mm r\kern0.15mm,\kern0.07mm 0\ar 1)\,,\kern0.15mm(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1))
= \{\kern0.15mm\{\,(\kern0.37mm r\kern0.15mm,\kern0.07mm 0\ar 1)\,,\kern0.15mm(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\}\,,\kern0.15mm
\{\,(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\}\kern0.15mm\} \KP1 $. \inskipline{.4}0
Since \math{u} is a function with \math{{}^{}{\rm rng}\,{}_{{}^{}} u\subseteq\ovbbR} there are \math{
x\kern0.37mm,\kern0.15mm s} with \inskipline{.4}{22}
$ \{\kern0.07mm\{\kern0.07mm\{\,0\ar 1\}\kern0.07mm\}\kern0.07mm\}
= \{\,(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\}
= (\kern0.37mm x\kern0.37mm,\kern0.15mm(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1))$ \inskipline{.2}{36.8}
${}= \{\kern0.15mm\{\,x\kern0.37mm,\kern0.15mm(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,
\}\,,\kern0.15mm\{\,(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,
\}\kern0.15mm\} \KP1 $, \inskipline{.4}0
and hence \mathss38{
\{\,x\kern0.37mm,\kern0.15mm(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\} =
\{\kern0.07mm\{\,0\ar 1\}\kern0.07mm\} = \{\,(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm
0\ar 1) \, \} }, \,whence further \mathss30{\{\,0\ar 1\} } \mathss08{{\KN{.99}}
= (\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)
= \{\kern0.15mm\{\,(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1)\,,\kern0.15mm(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\}\,,\kern0.15mm
\{\,(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\}\kern0.15mm\} }. Then we get \inskipline{.2}{7.7}
$0\ar 1\in\{\,0\ar 1\} \in \{\kern0.07mm\{\,0\ar 1\}\kern0.07mm\} \in
\{\kern0.07mm\{\kern0.07mm\{\,0\ar 1\}\kern0.07mm\}\kern0.07mm\} = \{\,(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\} =
0\ar 1 \kern0.37mm $, \,a {\sl contradiction\kern0.15mm}.
\end{proof} | 2,838 | 362,273 | en |
train | 0.47.15 | For \math{0<q<\lower1.05mm\hbox{$^+$}\infty} we assume that \math{ \lower1.05mm\hbox{$^+$}\infty\RHB{.25}{\KPt8^q}
= \lower1.05mm\hbox{$^+$}\infty} in the following
\begin{constructions}[of Lebesgue quasi\kern0.37mm-\kern0.37mm norms]\label{Ctr |x|_lL^p} $\null$ \inskipline{.7}2
(1) \ $\|\,x\,\|\lllnor_p=\uniqset s : [\KPp1.4 0<p<\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm
s=\big(\kern0.15mm\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\emath x} \kern0.37mm | \KP1
x\fvalss01 i\KP1|\suba\RHB{.25}{^p}\kern0.37mm\sbig)0\,^{p^{-1}} \KPp 1.4 \big ] \kern0.37mm $ \inskipline{.2}{40}
or $\kern0.37mm [\KPp1.4 p = \lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm s = \sup\kern0.15mm\big\{\,
|\,t\,|\suba\kern-0.3mm:t\in{}^{}{\rm rng}\,{}_{{}^{}} x\KPt8\} \KPp 1.4 \big ] \KP1 $, \inskipline{.5}2
(2) \ $\|\,x\,\|\Lnorss33^p_\mu=\uniqset s : [\KPp1.4 0<p<\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm \label{ctr L^p-norm}
\aall{}^{}\Cal Omega\,{}^{}\Cal Omega = \bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\impss30{}$ \inskipline{.2}{35.8}
$s = \inf\kern0.15mm\big\{\,\big(\kern0.15mm\int_{\KPp1.1{}^{}\Cal Omega\,}\varphi\rmdss11\mu\kern0.37mm)\KP1
^{p^{-1}}\kern-0.63mm:\varphi\in\kern0.15mm^{}^{}\Cal Omega\KP1[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9]\kern0.37mm$
and \inskipline{.2}{23.4}
$\varphi\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.07mm\barscTbb_R\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ and $\kern0.37mm
\aall{\eta\,,\kern0.15mm t}\,(\kern0.37mm\eta\kern0.37mm,\kern0.07mm t\kern0.37mm)\in x\impss33
|\,t\,|\suba\RHB{.25}{^p} \le \varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1\} \KPp 1.4 \big ] $ \inskipline{.2}{7.5}
or $\kern0.37mm[\KPp1.4 p=\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm s=\inf\,\{\,\smb M:\smb M\in
\rbb R^+\kern0.15mm$ and $\kern0.37mm\aall A\,\eexi N\,A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ \inskipline{.2}{23.5}
${}\impss03 N\in\mu\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.37mm$ and $\kern0.37mm\sup\kern0.15mm\big\{\,
|\,t\,|\suba\kern-0.3mm:t\in x\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.15mm A\setminus N\kern0.37mm)\KPt8\}\le\smb M\KP1\}
\KPp 1.4 \big ] \KP1 $.
\end{constructions}
For completeness' sake, in Constructions \ref{defi $L^p$} below of the
generalized Lebesgue\,--\,Bochner spaces we have included items (7) and (11)
where we define \math{\suptext{vc}0\Lrs03^p(\vcal Q\kern0.15mm) } and \mathss38{
\LLrs03^p(\kern0.37mm Q\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) }. There we utilize the concepts of
{\sl quasi\kern0.37mm-\kern0.15mm\esl Euclidean vector column\kern0.15mm} and
{\sl quasi\kern0.37mm-\kern0.15mm usual space\kern0.15mm}. To make matters precise, we give the
following
\begin{definitions}
(1) \ Say that \math{\vcolQ} is a {\it quasi\kern0.37mm-\kern0.15mm\eit Euclidean \mathss36{
\bosy K}--\,vector column\kern0.37mm} if{}f there are $Q\kern0.37mm,\kern0.15mm\Yps\kern0.07mm,\kern0.15mm\vPi\kern0.37mm$
with \math{\vcolQ = (\kern0.37mm Q\kern0.37mm,\kern0.15mm\Yps\kern0.07mmp,\kern0.07mm\vPi\kern0.37mm) } and such that \math{
Q\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\Yps} and \math{\Yps\in\roman{LCS}\kern0.4mmps5(\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R) } and $\vPi\in\tvsps0(K)$
hold with \math{{\rm dim_{_{\kern.2mm Ha}}}\Yps\in\mathbb No} and \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and for every
\math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi
} there is \math{
u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } with \math{\xi=\Bnull_\vPi} or \mathss06{
u\fvalss02\xi \not= 0 }. \inskipline{.5}2
(2) \ Say that \math{\ebit F} {\it usualizes} \math{F} over \math{\bosy K}
if{}f \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and there is \math{k\in\mathbb No} with \math{
(\kern0.37mm\emptyset\,,\kern0.07mm\bosy K\kern0.37mm)\,,\kern0.15mm(\kern0.37mm k\kern0.37mm,\kern0.07mm F\kern0.37mm)\in
\ebit F\in\kern0.15mm^{k\kern0.37mm +\kern0.37mm 1.}\,\hbox{\font\SweD =cmssbx10\SweD U}{} } and for every \math{i\in k} there are \math{
i\ar 1\kern0.37mm,\kern0.15mm i\ar 2\in i\kern0.37mm\lower1mm\hbox{$^{^+}$} } and \math{l\in\mathbb N} and \math{
\ebit E\in\kern0.15mm^l\,(\kern0.37mm\ebit F\KPt8\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} i\kern0.37mm\lower1mm\hbox{$^{^+}$})} with \mathss38{
\ebit F\fvalss61 i\kern0.37mm\lower1mm\hbox{$^{^+}$} \in \{\KPt8\bmii8 F\fvalss61 i\ar 1\sqcap\kern0.15mm
(\kern0.07mm\ebit F\fvalss61 i\ar 2\kern0.07mm)\,,\kern0.07mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K\text{\,-\kern0.15mm}
\prodsubtext{tvs}\ebit E\KP1\} }. \inskipline{.5}2
(3) \ Say that \math{F} is {\it quasi\kern0.37mm-\kern0.15mm usual\,} over \math{\bosy K}
if{}f \inskipline0{18.8}
there is \math{\ebit F} such that \math{\ebit F} usualizes \math{F} over \math{
\bosy K}.
\end{definitions}
A quasi\kern0.37mm-\kern0.15mm usual space necessarily has finite nonzero dimension. For
example \linebreak the space \math{ F =
\bosy R\kern0.15mm\sqcap(\kern0.07mm\bosy R\kern0.15mm\sqcap\bosy R\kern0.37mm)\expnota^\ssmb N]_{tvs}} is
quasi\kern0.37mm-\kern0.15mm usual over \math{\bosy R} when \math{ \bosy R \in \{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and $
\smb N\in\mathbb N\kern0.15mm\,$, with for example \math{
\seqss44{\bosy R\,,\kern0.15mm\bosy R\kern0.15mm\sqcap\bosy R\,,
(\kern0.07mm\bosy R\kern0.15mm\sqcap\bosy R\kern0.37mm)\expnota^\ssmb N]_{tvs}\kern0.07mm,\kern0.15mm F} }
usualizing \mathss31{F}.
\begin{lemma}\label{Le for q-usu}
For \PouN$\kern0.37mm\sbi{\iota\kern0.37mm=\kern0.37mm\sixroman{1\kern0.15mm,\kern0.37mm 2}}\,,$ let $\,
\varUpsilon\kern-0.63mm\sbi\iota\kern0.15mm$ be quasi\kern0.37mm-\kern0.15mm usual over $\kern0.37mm
\bosy K\kern-0.2mm\sbi\iota$ with $\kern0.37mm Q\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\varUpsilon\kern-0.63mm\sbi\iota\kern0.37mm$. If
also $\,[\ Q\not=\emptyset\kern0.15mm$ and $\,\bosy K\kern-0.3mm\ar 1=\bosy K\kern-0.2mm\ar 2\ ]$ or
$\,\Int_taurd{\varUpsilon_\iota} Q\not=\emptyset\,,$ \,then $\,
\varUpsilon\aar 1=\varUpsilon\aar 2\,$.
\end{lemma}
Thus for example quasi\kern0.37mm-\kern0.15mm usual spaces \math{\Yps} over \math{\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R} are
such that every single point \math{\eta\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\Yps} uniquely determines the
whole algebraic and topological structure \mathss30{\Yps}. The proof of
Lemma \ref{Le for q-usu} is given in \cite{Hif}\,. It is quite long and
requires delving in the set theoretic formal construction of the complex
number system starting from the set \math{\mathbb No} of natural numbers, and so we
omit it here.
For \math{Q\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\Yps} this allows us to define a structured vector space
\math{\roman S\,(\kern0.37mm Q\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) } based on a set of functions \math{
Q\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} without explicit reference to the structure \math{\Yps} by
putting \math{\roman S\,(\kern0.37mm Q\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) =
\roman S\,(\kern0.37mm Q\,\sbi\Yps\kern0.15mm,\kern0.07mm\vPi\kern0.37mm) } when the latter is already
defined. So we just get a bit simpler notation for the same space. | 4,058 | 362,273 | en |
train | 0.47.16 | (3) \ Say that \math{F} is {\it quasi\kern0.37mm-\kern0.15mm usual\,} over \math{\bosy K}
if{}f \inskipline0{18.8}
there is \math{\ebit F} such that \math{\ebit F} usualizes \math{F} over \math{
\bosy K}.
\end{definitions}
A quasi\kern0.37mm-\kern0.15mm usual space necessarily has finite nonzero dimension. For
example \linebreak the space \math{ F =
\bosy R\kern0.15mm\sqcap(\kern0.07mm\bosy R\kern0.15mm\sqcap\bosy R\kern0.37mm)\expnota^\ssmb N]_{tvs}} is
quasi\kern0.37mm-\kern0.15mm usual over \math{\bosy R} when \math{ \bosy R \in \{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and $
\smb N\in\mathbb N\kern0.15mm\,$, with for example \math{
\seqss44{\bosy R\,,\kern0.15mm\bosy R\kern0.15mm\sqcap\bosy R\,,
(\kern0.07mm\bosy R\kern0.15mm\sqcap\bosy R\kern0.37mm)\expnota^\ssmb N]_{tvs}\kern0.07mm,\kern0.15mm F} }
usualizing \mathss31{F}.
\begin{lemma}\label{Le for q-usu}
For \PouN$\kern0.37mm\sbi{\iota\kern0.37mm=\kern0.37mm\sixroman{1\kern0.15mm,\kern0.37mm 2}}\,,$ let $\,
\varUpsilon\kern-0.63mm\sbi\iota\kern0.15mm$ be quasi\kern0.37mm-\kern0.15mm usual over $\kern0.37mm
\bosy K\kern-0.2mm\sbi\iota$ with $\kern0.37mm Q\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\varUpsilon\kern-0.63mm\sbi\iota\kern0.37mm$. If
also $\,[\ Q\not=\emptyset\kern0.15mm$ and $\,\bosy K\kern-0.3mm\ar 1=\bosy K\kern-0.2mm\ar 2\ ]$ or
$\,\Int_taurd{\varUpsilon_\iota} Q\not=\emptyset\,,$ \,then $\,
\varUpsilon\aar 1=\varUpsilon\aar 2\,$.
\end{lemma}
Thus for example quasi\kern0.37mm-\kern0.15mm usual spaces \math{\Yps} over \math{\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R} are
such that every single point \math{\eta\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\Yps} uniquely determines the
whole algebraic and topological structure \mathss30{\Yps}. The proof of
Lemma \ref{Le for q-usu} is given in \cite{Hif}\,. It is quite long and
requires delving in the set theoretic formal construction of the complex
number system starting from the set \math{\mathbb No} of natural numbers, and so we
omit it here.
For \math{Q\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\Yps} this allows us to define a structured vector space
\math{\roman S\,(\kern0.37mm Q\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) } based on a set of functions \math{
Q\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} without explicit reference to the structure \math{\Yps} by
putting \math{\roman S\,(\kern0.37mm Q\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) =
\roman S\,(\kern0.37mm Q\,\sbi\Yps\kern0.15mm,\kern0.07mm\vPi\kern0.37mm) } when the latter is already
defined. So we just get a bit simpler notation for the same space.
\insubsubhead Suitable locally convex spaces \label{Ss suit lcs}
Suitable locally convex spaces are those that are obtained from some
\erm Banachable space by weakening the topology so that we {\sl do not\kern0.15mm}
get more bounded sets. Our basic important examples of suitable spaces are the
weak$^*$ duals \math{E\dlsigss22} of \erm Banachable spaces \mathss35{E}. We
put the following
\begin{definitions}\label{df suit}
Say that\inskipline{.5}2
(1) \ $\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm$ is a {\it dominating norm\kern0.37mm} for \math{E} if{}f
there is \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} with \inskipline09
$E\in\tvsps0(K)\kern0.37mm$ and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} a norm on \math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E} with \inskipline09
$\bouSet E = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\capss31\{\,B:\eexi{n\in\rbb Z^+}\,B \subseteq
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KP{1.1} 0\,,\kern0.07mm n\KPt9]\KP1\big\} \KP1 $, \inskipline{.5}2
(2) \ $E\kern0.37mm$ is {\it almost suitable\kern0.37mm} over \math{\bosy K} if{}f \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{E\in\roman{LCS}\kern0.4mmps0(K)} and \inskipline09
there is a normable \math{F\in\roman{LCS}\kern0.4mmps0(K)} with \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} and $\kern0.37mm
\bouSet E\subseteq\bouSet F\kern0.15mm$, \inskipline{.5}2
(3) \ $E\kern0.37mm$ is {\it suitable\kern0.37mm} over \math{\bosy K} if{}f \math{ \bosy K
\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{E\in\roman{LCS}\kern0.4mmps0(K)} and \inskipline09
there is \math{F\in\roman{BaS}\kern0.4mmps0(K)} with \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} and $\kern0.37mm
\bouSet E\subseteq\bouSet F\kern0.15mm$. \inskipline{.5}2
For \q{almost suitable} or \q{suitable} in place of \kern0.15mm X also say that \inskipline0{38.6}
$E\kern0.37mm$ is X if{}f \math{E} is X over \math{\bosy K} for some \mathss31{
\bosy K}.
\end{definitions}
If \math{ \bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{E\in\roman{LCS}\kern0.4mmps0(K)} and \math{F\in\roman{BaS}\kern0.4mmps0(K)}
with \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} and \ú$\kern0.37mm\bouSet E$ \linebreak
\ú${\kern-0.63mm}\subseteq\bouSet F\kern0.15mm$, then \math{
\bouSet E=\bouSet F} holds since from \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} we get \mathss35{
\bouSet F\subseteq\bouSet E}. If we \linebreak
also have \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm G\in\roman{BaS}\kern0.4mmps0(K)} and \mathss32{
\bouSet E\subseteq\bouSet G}, \,then \math{F=G} holds. This is seen by noting that
\erm Banachable spaces are bornological, and hence have the strongest locally
convex topology with the same bounded sets. Thus \math{F} is the unique
\erm Banachable space from which \math{E} is obtained by weakening the
topology. The dominating norms \linebreak
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm$ for \math{E} are precisely the
compatible norms for \mathss30{F}, \,and then \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm,\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm) } is a corresponding (\kern0.15mm norm{\sl ed\kern0.37mm})
Banach space. | 3,587 | 362,273 | en |
train | 0.47.17 | there is a normable \math{F\in\roman{LCS}\kern0.4mmps0(K)} with \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} and $\kern0.37mm
\bouSet E\subseteq\bouSet F\kern0.15mm$, \inskipline{.5}2
(3) \ $E\kern0.37mm$ is {\it suitable\kern0.37mm} over \math{\bosy K} if{}f \math{ \bosy K
\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{E\in\roman{LCS}\kern0.4mmps0(K)} and \inskipline09
there is \math{F\in\roman{BaS}\kern0.4mmps0(K)} with \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} and $\kern0.37mm
\bouSet E\subseteq\bouSet F\kern0.15mm$. \inskipline{.5}2
For \q{almost suitable} or \q{suitable} in place of \kern0.15mm X also say that \inskipline0{38.6}
$E\kern0.37mm$ is X if{}f \math{E} is X over \math{\bosy K} for some \mathss31{
\bosy K}.
\end{definitions}
If \math{ \bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{E\in\roman{LCS}\kern0.4mmps0(K)} and \math{F\in\roman{BaS}\kern0.4mmps0(K)}
with \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} and \ú$\kern0.37mm\bouSet E$ \linebreak
\ú${\kern-0.63mm}\subseteq\bouSet F\kern0.15mm$, then \math{
\bouSet E=\bouSet F} holds since from \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} we get \mathss35{
\bouSet F\subseteq\bouSet E}. If we \linebreak
also have \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm G\in\roman{BaS}\kern0.4mmps0(K)} and \mathss32{
\bouSet E\subseteq\bouSet G}, \,then \math{F=G} holds. This is seen by noting that
\erm Banachable spaces are bornological, and hence have the strongest locally
convex topology with the same bounded sets. Thus \math{F} is the unique
\erm Banachable space from which \math{E} is obtained by weakening the
topology. The dominating norms \linebreak
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm$ for \math{E} are precisely the
compatible norms for \mathss30{F}, \,and then \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm,\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm) } is a corresponding (\kern0.15mm norm{\sl ed\kern0.37mm})
Banach space.
One should observe that the bornology of a suitable space {\sl does not\kern0.15mm}
determine the dual, i.e.\ there exist suitable spaces obtained by weakening
the same \erm Banachable space but with different duals. This is seen by
considering \math{\ell\KPt8^1\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm)\subw0 } and \ú$\kern0.37mm
\ell\KPt8^1\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm)\subsigrs03$ \linebreak
which both are obtained by weakening \mathss38{
\ell\KPt8^1\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm) }. The former has the initial topolo- gical
vector structure from \math{(\kern0.37mm\roman I\,A\,,\kern-0.3mm\tvbbR4^A\kern0.15mm\big) } for \math{
A=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\hbox{\font\≈=cmmi12\≈c}\lower.8mm\hbox{\font\≈=cmr6\≈o}\kern.4mm(\kern0.37mm\mathbb No\kern0.07mm) } and the latter for \mathss30{ A = \kern-0.3mm} \mathss03{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm) } when we let \mathss39{ \roman I\,A =
\big\langle\big\langle\kern0.37mm\sum\KP1(\kern0.37mm x\cdot y\kern0.37mm) : y\in A\KP1\rangle :
x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\ell\KPt8^1\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm)\KP1\rangle }.
\begin{lemma}\label{Le suit dom}
Let $\,E$ be almost suitable with $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a dominating norm.
Then for
every $\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\in\Bqnorm E$ there is $\,\smb M\in\rbb R^+$ with $\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\le\smb M\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10 x\kern0.37mm)$ for all $\,
x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\,$.
\end{lemma}
\begin{proof} We have \mathss35{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb 20 I\in\bouSet E
}, \,and letting \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\in\Bqnorm E} then \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\,[\KP{1.1}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb 26 I\,] \in {\kern-0.63mm}} \mathss03{
\bouSet\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R } and hence for \math{ \smb M =
\sup\KPt8(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\,[\KP{1.1}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb 26 I\,]\kern0.37mm\sbig)0 }
we have \mathss35{\smb M < \lower1.05mm\hbox{$^+$}\infty }. Considering \ú$\kern0.37mm x\in {\kern-0.63mm}$ $
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\,$, if \math{x=\Bnull_E} holds, we trivially have \mathss38{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x=0\le 0=\smb M\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10 x\kern0.37mm) }. Otherwise
taking \math{\smb A=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10 x} we have \mathss34{
(\kern0.15mm\smb A\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1\,}x\kern0.37mm)\svs E \in \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb 20 I}, \,and
hence \mathss30{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm\smb A\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1\,}x\kern0.37mm)\svs E \le
\smb M} and further \mathss38{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x \le \smb M\,\smb A =
\smb M\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10 x\kern0.37mm) }.
\end{proof} | 3,815 | 362,273 | en |
train | 0.47.18 | \begin{lemma}\label{Le E'' adher}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\roman{LCS}\kern0.4mmps0(K)$ and \ú$\, w \in
\Cal L\,(\kern0.15mm E\dlbetss12\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) \KP1 $. Then there is some \ú$\,
B\in\bouSet E$ such that for every finite \ú$\, A \subseteq
\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) $ there is \ú$\, x \in B $ with $\,
|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x - w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u\KP1|\suba\le 1 $ for all $\,u\in A\,$.
\end{lemma}
\begin{proof} Putting \mathss38{D\ar 1=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K\capss21\{\,t:|\,t\,|\suba
\le 1 \KPt8\} }, \,from \math{ w \in
\Cal L\,(\kern0.15mm E\dlbetss12\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } we first get existence of
some nonempty absolutely convex bounded set \math{B} in \math{E} such that for \mathss03{
U = \Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) \capss31 \{\, u : u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm B
\subseteq D\ar 1\kern0.37mm\} } we have \mathss34{w\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm U \subseteq D\ar 1 }. Then
for the canonical evaluation \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \seqss33{
\roman{ev}\sbi{\kern0.15mm\emath x}\,|\KP1\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) :
x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} } and for \math{\scrmt T=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\dlbetss12{\kern-0.63mm}\dlsigss02) }
from the {\sl bipolar theorem\kern0.15mm} \cite[3.3.1\kern0.15mm, p.\ 192]{Ho} or
\cite[8.2.2\kern0.37mm, p.\ 149]{Jr} we see \math{ w \in
\roman{Cl}\sbi{\KPt8\scrm7 T\KP1}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm B\kern0.37mm) } to hold whence
the assertion follows.
\end{proof}
The content of \cite[Lemma 8.17.8 \erm B\kern0.37mm, p.\ 585]{Edw} is in the
following
\begin{lemma}\label{Le 8.17.8 B}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\roman{LCS}\kern0.4mmps0(K)$ be normable{\kern0.37mm\rm, }and
let \ú$\,F=E\dlbetss12\,$. Also let $\,S\ar 1$ be a linear subspace in $\,
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F$ such that \ú$\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 S\ar 1$ is a separable
topology{\kern0.37mm\rm, }and let \ú$\, w \in
\Cal L\,(\kern0.15mm F_{\kern0.15mm/\kern0.37mm\aars S_1},\kern0.07mm\bosy K\kern0.37mm) \,$. Then there is \ú$\,
\bosy x\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E$ with \ú$\,{}^{}{\rm rng}\,{}_{{}^{}}\bosy x\in\bouSet E$ and such that $\,
w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u = \lim\,(\kern0.37mm u\circss01\bosy x\kern0.37mm)$ holds for every $\,u\in
S\ar 1\kern0.37mm$.
\end{lemma}
\begin{proof} By Hahn\,--\,Banach there is \math{\bar w\in
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) } with \mathss35{w\subseteq\bar w}.
Furthermore, for \linebreak
\mathss03{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2} the canonical embedding \mathss36{E\to
E\dlbetss12{\!}\dlbetss02}, \,i.e.\ for \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2 =
\seqss33{\roman{ev}\sbi{\kern0.15mm\emath x}\,|\KP1\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F:x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} } and for \mathss03{
\scrmt T=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\dlbetss12{\!}\dlsigss04) } by \kern0.15mm Lemma \ref{Le E'' adher}
above, there is some \math{B\in\bouSet E} such that \mathss03{\bar w\in B\ar 2}
holds for \mathss38{B\ar 2 =
\roman{Cl}\sbi{\KPt8\scrm7 T\KPt8}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2\kern-0.3mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm B\kern0.37mm) }. Now
letting \math{\scrmt T\aR 1} be the initial topology from \mathss30{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm \kern0.15mm F_{\kern0.15mm/\kern0.37mm\aars S_1}\sbig)0\dlsigss12\kern0.07mm) } under \math{
\seqss33{z\KP1|\KP1 S\ar 1\kern-0.3mm:z\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)} } we
have \math{\scrmt T\aR 1\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss02 B\ar 2\subseteq \scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 B\ar 2 } with \math{
\scrmt T\aR 1\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss02 B\ar 2 } semimet- rizable. Hence there is \math{\bosy x
\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,B} with \mathss30{w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u = \bar w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u =
\lim\,(\kern0.37mm u\circss01\bosy x\kern0.37mm) } for \mathss34{u\in S\ar 1}.
\end{proof}
\begin{lemma}\label{Le qtvs}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\tvsps0(K)$ and let $\,S$ be a vector
subspace in $\,\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\,$.
Also let \ú$\,F=E\,/\tvsquotient S$ and \ú$\,
\Cal V = \{\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,\smb X:U\capss02\smb X\not=\emptyset\KP1\} :
U\kern-0.3mm\in\Cal U\KP1\}$ \linebreak
where $\,\Cal U$ is a filter base for $\,\neiBoo E\,$.
Then $\,\Cal V$ is a filter base for $\,\neiBoo F\kern0.15mm$.
\end{lemma}
\begin{proof} With \mathss38{ \tweq = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F \capss31 \{\,
(\kern0.37mm x\kern0.37mm,\kern0.07mm\smb X\kern0.15mm):x\in\smb X\,\} } we know from the discussion in
\cite[p.\ 104]{Ho} that \math{\tweq} is continuous and open \mathss30{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\to\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}, \,and consequently \linebreak
\ú$\Cal V=\tweq\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.15mm\Cal U\subseteq
\neiBoo F\kern0.37mm$ holds. Moreover, for every \math{V\in\neiBoo F} we have \ú$\kern0.37mm
\tweq\invss64\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm V\in\neiBoo E$ \linebreak
and hence there is \math{U\in\Cal U}
with $\kern0.37mm U\subseteq\tweq\invss64\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm V\kern0.07mm$, but then \math{
\tweq\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} U\subseteq V} holds.
\end{proof} | 4,057 | 362,273 | en |
train | 0.47.19 | \Ssubhead A Measurability and integration \label{Sec A}
In this section, we first explain what it means for functions \math{ x :
{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} to be measurable when \math{\vPi} is a topological vector
space and \math{{}^{}\Cal Omega} is a set equipped with a positive measure \mathss36{
\mu}. In the next section, for \math{0\le p\le\lower1.05mm\hbox{$^+$}\infty} we construct the
spaces $\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm$ and \math{
\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } of certain equivalence classes \math{
\smb X} of such \mathss34{x}.
By saying that \math{\mu} is a {\it positive measure\kern0.37mm} on \math{{}^{}\Cal Omega} we
mean that \math{\mu} is a function with \math{\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu={}^{}\Cal Omega} and \math{
{}^{}{\rm rng}\,{}_{{}^{}}\mu\subseteq[\KP1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } and \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} a
\rsigma3algebra and such that \linebreak
\ú$ \sum\,(\kern0.37mm\mu\KP1|\KP1\scrmt A\kern0.37mm) =
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\big(\kern0.15mm\bigcup\,\scrmt A\kern0.37mm) \kern0.37mm$ holds for any countable
disjoint \mathss36{\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}. Here we assume that the definitions
associated with sum conventions are arranged so that \linebreak
\ú$ \sum\,\emptyset = 0\kern0.37mm$
holds. Further, by a \rsigma0{\it algebra\kern0.37mm}, usually written
\q{$\sigma\,$-\,algebra}, we mean any \linebreak
$\scrmt A\kern0.37mm$ such that \math{
\bigcup\,\scrmt A\kern0.15mm\setminus\kern0.37mm\bigcup\,\scrmt B\in\scrmt A} holds for any
countable \mathss36{\scrmt B\subseteq\scrmt A}. A positive measure $\mu\kern0.37mm$ is
\rsigma1{\it finite\kern0.37mm} if{}f \math{
\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\subseteq\bigcup\,\scrmt A } holds for some countable \mathss35{
\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+ }.
To compact language in some, quite rare cases, we introduce the concept of
measure space as follows. Say that \math{P} is a {\it measure space\kern0.37mm} if{}f
there are \math{\mu} and \math{{}^{}\Cal Omega} such \linebreak
that \math{\mu} is a positive
measure on \math{{}^{}\Cal Omega} with \mathss38{P = (\kern0.37mm{}^{}\Cal Omega\kern0.37mm,\kern0.07mm\mu\kern0.37mm) }. We
also say that \math{\mu} is a positive measure if{}f \math{\mu} is a positive
measure on \mathss36{\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}, \,and a measure space \math{P} we \linebreak
say
to be \rsigma6finite in the case where \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm P} is such. \vskip.5mm
\insubsubhead Measurability of measure\kern0.37mm-\kern0.15mm vector maps \label{Ss C1}
We consider {\it mv\kern0.37mm-\kern0.15mm map\,}s, short for \q{measure\kern0.37mm-\kern0.15mm vector},
which are triplets, i.e.\ ordered pairs \math{ \tilde x =
(\kern0.37mm x\kern0.37mm,\kern0.07mm\varXi\kern0.37mm) = (\kern0.37mm x\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } where in
turn \math{\varXi = (\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is an {\it mv\kern0.37mm-\kern0.15mm pair\kern0.15mm}.
This means that \math{\vPi} is a real or complex topological vector space and \math{
\mu} is a positive measure on some \math{{}^{}\Cal Omega} and \math{ x : {}^{}\Cal Omega \to
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} is a function. In order to introduce some concepts of
measurability for such mv\kern0.37mm-\kern0.15mm maps we first put the following
\begin{definitions}\label{df simple}
(1) \ Say that \math{\sigma} is {\it simple\kern0.37mm} in \math{\varXi} if{}f
there are \math{\bosy K\kern0.15mm,\kern0.15mm\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm\vPi} with \math{\mu}
a po- sitive
measure on \math{{}^{}\Cal Omega} and \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and
\math{\vPi\in\tvsps0(K)} and
\math{\varXi=(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) }
and \mathss03{\sigma\in\kern0.15mm^{}^{}\Cal Omega\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} and
\math{{}^{}{\rm rng}\,{}_{{}^{}}\sigma} finite and
\math{\{\,\sigma\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm\xi\kern0.37mm\}:\xi\in
{}^{}{\rm rng}\,{}_{{}^{}}\sigma\kern0.07mm\setminus\{\,\Bnull_\vPi\}\kern0.15mm\}\subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+}, \inskipline{.5}2
(2) \ Say that \math{\bosy\sigma} is a {\it simple sequence\kern0.37mm} in \math{\varXi}
if{}f \math{\bosy\sigma\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\hbox{\font\SweD =cmssbx10\SweD U}{}} and \inskipline0{41.5}
$\sigma\kern0.37mm$ is simple in \math{\varXi} for all \math{\sigma\in
{}^{}{\rm rng}\,{}_{{}^{}}\bosy\sigma}.
\end{definitions}
Let \math{\text{\efss R\KPt8}\tilde x\,A=
\uniqset\tilde z:\eexi{x\kern0.37mm,\kern0.15mm\mu\,,\kern0.15mm\vPi}\,
\tilde x=(\kern0.37mm x\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \mathss38{
\tilde z=(\kern0.37mm x\KP1|\KP1 A\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\kern0.37mm) }.
\begin{def:al schemas}\label{df meas}
For any mv\kern0.37mm-\kern0.15mm map \math{\tilde x=
(\kern0.37mm x\kern0.37mm,\kern0.07mm\varXi\kern0.37mm)=(\kern0.37mm x\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } with \math{
{}^{}\Cal Omega={{}^{}{\rm dom}\,{}_{{}^{}}} x}
assuming $\vPi\in\tvsps0(K)$ , first say that \inskipline12
(1) \ $\tilde x\kern0.37mm$ is {\it measurable\kern0.37mm} if{}f \math{
\{\,x\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm U\kern-0.3mm:U\in\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\KP1\}\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} holds, \inskipline{.5}2
(2) \ $\tilde x\kern0.37mm$ is {\it simply measurable\kern0.37mm} if{}f there is \math{
\bosy\sigma} with \math{\bosy\sigma} a simple sequence in \math{\varXi} \inskipline09
and \math{\bosy\sigma\to x} in top \mathss35{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.37mm]_{ti} }, $
\bosy\sigma\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\kern0.37mm\big(\,^{}^{}\Cal Omega\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.37mm) $ \KP6 \inskipline{.5}2
(3) \ $\tilde x\kern0.37mm$ is {\it scalarly measurable\kern0.37mm} if{}f
$(\kern0.37mm u\circ x\,;\kern0.15mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ is measurable for all
$u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) \,$.\KN9 \inskipline10
Then for any of \q{measurable}, \q{simply measurable} or
\q{scalarly measurable} in place of \,X\kern0.37mm, say that \inskipline12
(4) \ $\tilde x\kern0.37mm$ is {\it almost\kern0.37mm} X if{}f
\math{\text{\efss R\KPt8}\tilde x\KP1(\kern0.37mm{}^{}\Cal Omega\kern0.15mm\setminus\kern0.15mm N\kern0.37mm) } is X
for some \mathss39{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \inskipline{.5}2
(5) \ $\tilde x\kern0.37mm$ is {\it finitely\kern0.37mm} X if{}f
\math{\text{\efss R\KPt8}\tilde x\,A} is X for
every \mathss36{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+}, \inskipline{.5}2
(6) \ $\tilde x\kern0.37mm$ is {\it finitely almost\kern0.37mm} X if{}f
\math{\text{\efss R\KPt8}\tilde x\,A} is almost X for
every \mathss36{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+}.
\end{def:al schemas}
In loose speach, we may express the content of \ref{df meas}\,(2) by saying
that \math{\bosy\sigma} is a {\sl sequence of simple functions converging
pointwise\kern0.15mm} to \mathss34{x}. Then for the \math{\sigma\in\bosy\sigma} there \linebreak
we may also say that \math{(\kern0.37mm\sigma\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is a
{\it simple\kern0.37mm} mv\kern0.37mm-\kern0.15mm map, and we may loosely say that \linebreak
$\sigma\kern0.37mm$ is a
{\sl simple function\kern0.15mm}. | 4,010 | 362,273 | en |
train | 0.47.20 | \begin{def:al schemas}\label{df meas}
For any mv\kern0.37mm-\kern0.15mm map \math{\tilde x=
(\kern0.37mm x\kern0.37mm,\kern0.07mm\varXi\kern0.37mm)=(\kern0.37mm x\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } with \math{
{}^{}\Cal Omega={{}^{}{\rm dom}\,{}_{{}^{}}} x}
assuming $\vPi\in\tvsps0(K)$ , first say that \inskipline12
(1) \ $\tilde x\kern0.37mm$ is {\it measurable\kern0.37mm} if{}f \math{
\{\,x\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm U\kern-0.3mm:U\in\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\KP1\}\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} holds, \inskipline{.5}2
(2) \ $\tilde x\kern0.37mm$ is {\it simply measurable\kern0.37mm} if{}f there is \math{
\bosy\sigma} with \math{\bosy\sigma} a simple sequence in \math{\varXi} \inskipline09
and \math{\bosy\sigma\to x} in top \mathss35{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.37mm]_{ti} }, $
\bosy\sigma\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\kern0.37mm\big(\,^{}^{}\Cal Omega\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.37mm) $ \KP6 \inskipline{.5}2
(3) \ $\tilde x\kern0.37mm$ is {\it scalarly measurable\kern0.37mm} if{}f
$(\kern0.37mm u\circ x\,;\kern0.15mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ is measurable for all
$u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) \,$.\KN9 \inskipline10
Then for any of \q{measurable}, \q{simply measurable} or
\q{scalarly measurable} in place of \,X\kern0.37mm, say that \inskipline12
(4) \ $\tilde x\kern0.37mm$ is {\it almost\kern0.37mm} X if{}f
\math{\text{\efss R\KPt8}\tilde x\KP1(\kern0.37mm{}^{}\Cal Omega\kern0.15mm\setminus\kern0.15mm N\kern0.37mm) } is X
for some \mathss39{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \inskipline{.5}2
(5) \ $\tilde x\kern0.37mm$ is {\it finitely\kern0.37mm} X if{}f
\math{\text{\efss R\KPt8}\tilde x\,A} is X for
every \mathss36{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+}, \inskipline{.5}2
(6) \ $\tilde x\kern0.37mm$ is {\it finitely almost\kern0.37mm} X if{}f
\math{\text{\efss R\KPt8}\tilde x\,A} is almost X for
every \mathss36{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+}.
\end{def:al schemas}
In loose speach, we may express the content of \ref{df meas}\,(2) by saying
that \math{\bosy\sigma} is a {\sl sequence of simple functions converging
pointwise\kern0.15mm} to \mathss34{x}. Then for the \math{\sigma\in\bosy\sigma} there \linebreak
we may also say that \math{(\kern0.37mm\sigma\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is a
{\it simple\kern0.37mm} mv\kern0.37mm-\kern0.15mm map, and we may loosely say that \linebreak
$\sigma\kern0.37mm$ is a
{\sl simple function\kern0.15mm}.
Note that by our definitions above we may also say e.g.\ that \math{\tilde x}
is {\sl measurable\kern0.15mm} if{}f there are \math{\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm\vPi\kern0.15mm,\kern0.15mm
x} with \math{{}^{}\Cal Omega\times\kern-0.2mm\{\,\Bnull_\vPi\} } simple in \math{
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \mathss03{ \tilde x =
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{x\in\kern0.15mm^{}^{}\Cal Omega\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } and \mathss36{
x\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.15mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}.
\begin{proposition}\label{pro-mea-equ}
Let \PouN$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}${\,\rm, }and let \PouN$\,\vPi\in\roman{LCS}\kern0.4mmps0(K)$ be
normable with $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ a separable topology. Also let $\,\mu$ be a \kern0.15mm
\rsigma1finite positive measure on $\,{}^{}\Cal Omega\,$. If in addition $ \tilde x =
(\kern0.37mm x\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with $\, x \in\kern0.15mm^{}^{}\Cal Omega\,
\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) \KP1 ${\rm, \,}then $\
(1) \equivss22 (2) \equivss22 (3) $ \,where {\rm \inskipline{.9}4
(1)} \ $(\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.07mm x\,;\kern0.15mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ is
measurable for all $\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi${\kern0.15mm\rm, \inskipline{.4}4
(2)} \ $\tilde x$ is simply measurable{\kern0.37mm\rm, \inskipline{.4}4
(3)} \ $\tilde x$ is measurable\kern0.37mm. \end{proposition}
\begin{proof} Since \math{ \{\KP1 \roman{ev}\kern0.15mm\sbi\xi \, | \KP1
\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) : \xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\KP1\} \subseteq
\Cal L\,(\kern0.07mm\vPi\dualsigma0\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } holds, we trivially have
the implication \mathss36{(3)\impss22(1)}. Likewise, we trivially have \mathss36{
(2)\impss22(1)}. It now suffices to prove that the implications \math{
(1)\impss22(3)} and \math{(3)\impss22(2)} hold. | 2,766 | 362,273 | en |
train | 0.47.21 | \end{def:al schemas}
In loose speach, we may express the content of \ref{df meas}\,(2) by saying
that \math{\bosy\sigma} is a {\sl sequence of simple functions converging
pointwise\kern0.15mm} to \mathss34{x}. Then for the \math{\sigma\in\bosy\sigma} there \linebreak
we may also say that \math{(\kern0.37mm\sigma\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is a
{\it simple\kern0.37mm} mv\kern0.37mm-\kern0.15mm map, and we may loosely say that \linebreak
$\sigma\kern0.37mm$ is a
{\sl simple function\kern0.15mm}.
Note that by our definitions above we may also say e.g.\ that \math{\tilde x}
is {\sl measurable\kern0.15mm} if{}f there are \math{\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm\vPi\kern0.15mm,\kern0.15mm
x} with \math{{}^{}\Cal Omega\times\kern-0.2mm\{\,\Bnull_\vPi\} } simple in \math{
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \mathss03{ \tilde x =
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{x\in\kern0.15mm^{}^{}\Cal Omega\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } and \mathss36{
x\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.15mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}.
\begin{proposition}\label{pro-mea-equ}
Let \PouN$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}${\,\rm, }and let \PouN$\,\vPi\in\roman{LCS}\kern0.4mmps0(K)$ be
normable with $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ a separable topology. Also let $\,\mu$ be a \kern0.15mm
\rsigma1finite positive measure on $\,{}^{}\Cal Omega\,$. If in addition $ \tilde x =
(\kern0.37mm x\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with $\, x \in\kern0.15mm^{}^{}\Cal Omega\,
\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) \KP1 ${\rm, \,}then $\
(1) \equivss22 (2) \equivss22 (3) $ \,where {\rm \inskipline{.9}4
(1)} \ $(\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.07mm x\,;\kern0.15mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ is
measurable for all $\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi${\kern0.15mm\rm, \inskipline{.4}4
(2)} \ $\tilde x$ is simply measurable{\kern0.37mm\rm, \inskipline{.4}4
(3)} \ $\tilde x$ is measurable\kern0.37mm. \end{proposition}
\begin{proof} Since \math{ \{\KP1 \roman{ev}\kern0.15mm\sbi\xi \, | \KP1
\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) : \xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\KP1\} \subseteq
\Cal L\,(\kern0.07mm\vPi\dualsigma0\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } holds, we trivially have
the implication \mathss36{(3)\impss22(1)}. Likewise, we trivially have \mathss36{
(2)\impss22(1)}. It now suffices to prove that the implications \math{
(1)\impss22(3)} and \math{(3)\impss22(2)} hold.
For \mathss36{(1)\impss22(3)}, letting \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} be some compatible norm for \mathss33{
\vPi}, \,let \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aar 1} be the corresponding dual norm, i.e.\ put \mathss39{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1=\seqss44{
\sup\KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)
:u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)} }.
\newline$\null
$
For \mathss36{
i\in\mathbb No}, \,then put \mathss38{ \roman B\,i =
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aar 1\!\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KP1 0\,,\kern0.15mm i\,\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,] }. Assuming
(1)\,, if
\linebreak
now \mathss37{U\in\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.07mm\vPi\dlsigss00\kern0.07mm) }, \,for any
fixed \math{i\in\mathbb No} and for \mathss35{\roman U\,i=U\capss14\roman B\,i }, \,
it suffices to prove that \math{x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\roman U\,i\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}
holds. For \mathss35{ \scriptm9 T\ar 1 =
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.07mm\vPi\dlsigss00\kern0.07mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33\roman B\,i }, \,now \math{
\scriptm9 T\aR 1} is the uniform $\kern0.37mm\scriptm{10}U\,$-- \linebreak
topology where \math{\scriptm{10}U } is the uniformity generated by \PouN$\kern0.37mm
\{\KP1\roman V\,\xi\,n:\xi\in D\kern0.37mm\text{ and }\kern0.37mm n\in\rbb Z^+\,\}$ where \math{
\roman V\,\xi\,n = \roman B\,i\times\roman B\,i \capss31 \{\,
(\kern0.37mm u\kern0.37mm,\kern0.07mm v\kern0.37mm) :
|\KP1 u\fvalss01\xi - v\fvalss01\xi\KP1| < n^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,\} } and \math{
D} is any countable $
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22(\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss43\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPt9 0\,,1\KPt8]\kern0.37mm)\,$--\,dense
set. By Alaoglu's theorem, we have \math{\scriptm9 T\aR 1} a compact topology.
Since \math{\scriptm{10}U } is generated by a countable set, we see that there
is some \mathss33{\scriptm9 T\aR 1}-- dense and countable set \mathss34{
D\aar 1}. Using (1) and noting that \
$x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.37mm\roman V\,\xi\,n\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\kern0.15mm u\kern0.15mm\}\kern0.15mm) = {} $ \vskip.4mm\centerline{$
\bigcap\KP1\{\,\{\KP1 t:|\KP1 x\fvalss01 t\fvalss21\xi -
u\fvalss01\xi\KP1| < n^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}
\kern0.37mm\text{ and }\,
|\KP1 x\fvalss01 t\fvalss21\zeta\KP1|\le i\,\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,\} :
\zeta\in D\KP1\}\KP1$,} \inskipline{.4}0
we see that \math{
x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.37mm\roman V\,\xi\,n\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\kern0.15mm u\kern0.15mm\}\kern0.15mm)\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} holds
when \mathss31{ (\kern0.37mm\xi\,,\kern0.07mm u\kern0.37mm,\kern0.07mm n\kern0.37mm) \in
D\times D\ar 1\kern-0.3mm\times\kern0.15mm\rbb Z^+}. It is left as an exercise to the reader
to show that \math{\roman U\,i} can be expressed as a union of finite
intersections of the sets \mathss37{\roman V\,\xi\,n\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\kern0.15mm u\kern0.15mm\} }.
Then \math{x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\roman U\,i\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} follows. | 3,738 | 362,273 | en |
train | 0.47.22 | \bigcap\KP1\{\,\{\KP1 t:|\KP1 x\fvalss01 t\fvalss21\xi -
u\fvalss01\xi\KP1| < n^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}
\kern0.37mm\text{ and }\,
|\KP1 x\fvalss01 t\fvalss21\zeta\KP1|\le i\,\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,\} :
\zeta\in D\KP1\}\KP1$,} \inskipline{.4}0
we see that \math{
x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.37mm\roman V\,\xi\,n\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\kern0.15mm u\kern0.15mm\}\kern0.15mm)\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} holds
when \mathss31{ (\kern0.37mm\xi\,,\kern0.07mm u\kern0.37mm,\kern0.07mm n\kern0.37mm) \in
D\times D\ar 1\kern-0.3mm\times\kern0.15mm\rbb Z^+}. It is left as an exercise to the reader
to show that \math{\roman U\,i} can be expressed as a union of finite
intersections of the sets \mathss37{\roman V\,\xi\,n\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\kern0.15mm u\kern0.15mm\} }.
Then \math{x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\roman U\,i\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} follows.
For \mathss36{(3)\impss22(2)}, we need the assumption that \math{\mu} be
\rsigma6finite. It is an easy exercise to show that if the implication to be
established holds for bounded measures, then it holds also for \rsigma6finite
ones. So we assume that \mathss36{\mu\fvalss01{}^{}\Cal Omega < \lower1.05mm\hbox{$^+$}\infty}. Further,
if we can show that for any fixed \mathss37{i\in\mathbb No}, \,and for \math{\bar x
=x\capss33(\kern0.37mm\hbox{\font\SweD =cmssbx10\SweD U}{}\times\roman B\,i\kern0.37mm)} with \mathss34{B={{}^{}{\rm dom}\,{}_{{}^{}}}\bar x}, \,
the required implication holds for \math{
(\kern0.37mm\bar x\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\vPi\dlsigss00\kern0.07mm)} in place of \mathss34{
\tilde x}, \,then it easily follows also for \mathss34{\tilde x}, observing
that \mathss36{ B = \bigcap \KP1 \{\, \{ \KP1 t :
|\KP1 x\fvalss01 t\fvalss21\zeta\KP1|\le i\,\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,\} :
\zeta\in D\KP1\}\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}.
As just explained, for \mathss36{(3)\impss22(2)}, assuming (3) and making the
additional assumptions that \math{\mu\fvalss01{}^{}\Cal Omega < \lower1.05mm\hbox{$^+$}\infty} and \math{
{}^{}{\rm rng}\,{}_{{}^{}} x\subseteq B\ar 1=\roman B\,i\ar 0 } for some fixed \mathss36{i\ar 0\in\mathbb No},
we should establish (2)\kern0.37mm. For this, we construct \math{ \bosy s \in
\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\kern0.37mm\big(\,^{}^{}\Cal Omega\kern0.37mm B\ar 1\kern0.07mm) } with \math{
(\kern0.37mm\smb S\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } a simple mv\kern0.37mm-\kern0.37mm map for
every \mathss36{\smb S\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy s}, \,and such that \math{\bosy s\to x} in
top \math{\scriptm9 T\expnota^\KPt8{}^{}\Cal Omega\kern0.15mm]_{ti} } when we take \linebreak \PouN$
\scriptm9 T=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.07mm\vPi\dlsigss00\kern0.07mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 B\ar 1\,$. Since \math{
\scriptm9 T} is a compact topology, and also the uniform $\kern0.37mm\scriptm{10}U\,
$-- topology with \math{\scriptm{10}U} being countably generated, we may first
choose some decreasing \linebreak \PouN$
\bosy w:\mathbb No\to\scriptm{10}U\kern0.37mm$ with \math{{}^{}{\rm rng}\,{}_{{}^{}}\bosy w} a symmetric base
for \mathss36{\scriptm{10}U}, \,and then some \math{\bosy u:\mathbb No\to\hbox{\font\SweD =cmssbx10\SweD U}{}}
with the following property. For every \math{i\in\mathbb No} there is \math{
k\in\mathbb No} with \math{\bosy u\fvalss01 i\in B\ar 1\kern0.15mm^k } and $ B\ar 1 \subseteq
\bigcup\KP1\{\,\bosy w\fvalss01 i\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\,\bosy u\fvalss01 i\fvalss21 j\,\}
:j\in k\,\}\KPt8$. For short writing \inskipline{.8}{6.75}
$\roman U\,i\,j =
\bosy w\fvalss01 i\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\,\bosy u\fvalss01 i\fvalss21 j\,\}
\kern0.15mm\setminus\, \bigcup \KPt8 \{\,
\bosy w\fvalss01 i\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\,\bosy u\fvalss01 i\fvalss21 l\,\}
: l\in j\,\} $ \ and \inskipline{.4}{6.75}
$\roman A\,i\,j = x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.37mm\roman U\,i\,j\,$, \,and taking \vskip.4mm
$ \null
\bosy s = \big\langle\, \bigcup \KPt8 \{\,
\roman A\,i\,j\times\{\,\bosy u\fvalss01 i\fvalss21 j\,\} :
\bosy u\kern0.37mm,\kern0.07mm i : j\in{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.37mm\bosy u\fvalss01 i\kern0.37mm)\,\} :
i\in\mathbb No \,\big\rangle \KP1$, \,we are done, \inskipline{.5}0
leaving the required straightforward verifications as exercises to the reader.
\end{proof} | 2,461 | 362,273 | en |
train | 0.47.23 | \begin{example}
Without separability of the topology \mathss30{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}, \,the implication \mathss30{
(1)\impss11(2)} \linebreak
in Proposition \ref{pro-mea-equ} need not hold. Indeed, with \math{
1<p<\lower1.05mm\hbox{$^+$}\infty} and \math{\vPi=\lll^p(\ssbb44 I) } and letting \math{\mu} be
the Lebesgue measure defined for all Lebesgue measurable sets \mathss35{A\subseteq
\bbI}, \,taking \math{\tilde x=(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) }
where \mathss38{x = \seqss34{\roman{ev\kern0.15mm}\sbi t\,|\KP1\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern-0.2mm:t\in\bbI}
}, \,we trivially have (1) since \linebreak
\mathss02{\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.07mm x\fvalss01 t =
x\fvalss01 t\fvalss21\xi = \xi\fvalss11 t }, \,and so \math{
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.07mm x\fvalss01 t \not = 0 } only for countably many \math{
t\in\bbI} for \linebreak
each fixed \mathss30{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }. However \math{\tilde x}
cannot be simply measurable since otherwise there would exist some countable
set \math{N\aar 0\subseteq\bbI} such that \math{x\fvalss01 t\fvalss21\xi\not=0 }
holds only for vec- \linebreak
tors \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} with \math{
\xi\invss46[\KP{1.2}\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.15mm\setminus\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\KP1]\capss42 N\aar 0
\not=\emptyset} when \mathss34{t\in\bbI}.
\end{example}
\begin{proposition}\label{Pro rfx si mea}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ be reflexive{\kern0.15mm\rm, }
and let \inskipline06
$(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\subsigrs03\kern0.07mm)$ be simply measurable. Then $\,
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ is simply measurable.
\end{proposition}
\begin{proof} Putting \mathss35{{}^{}\Cal Omega=\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}, \,let \math{
\bosy\sigma} be some simple sequence in \math{
(\kern0.37mm\mu\,,\kern0.07mm\vPi\subsigrs03\kern0.07mm) } with \math{ \bosy\sigma\to x} in top \mathss35{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\subsigrs03\kern0.07mm)\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.37mm]_{ti} }, \,and let \math{
S} be the closed linear span of \math{{}^{}{\rm rng}\,{}_{{}^{}}\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bosy\sigma} in \mathss31{
\vPi}. Then trivially \math{(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is scalarly
measurable, and by Hahn\,--\,Banach also \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 S} is a
separable topology with \mathss34{{}^{}{\rm rng}\,{}_{{}^{}} x\subseteq S}. Consequently by
Pettis' theorem the assertion follows.
\end{proof}
\begin{lemma}\label{Le Nu_1 ci y meas}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{LCS}\kern0.4mmps0(K)$ be normable with $\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a compatible norm and $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ a separable topology. Also let \vskip.5mm\centerline{$
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1 = \seqss44{ \sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I) : u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)} $} \inskipline{.5}0
and let $\,(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ be scalarly
measurable. Then $\,
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\,;\kern0.07mm\mu\,,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)$ is measurable.
\end{lemma}
\begin{proof} Putting \math{{}^{}\Cal Omega=\kern0.15mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and
taking any countable \math{D} such that \math{D}
is
\mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)}--\,dense,
for every \math{\eta\in{}^{}\Cal Omega} we
have \mathss38{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta= \label{Nu_1 = sup ...}
\sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm(\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)
=\sup\KPt8\{\KPt8\Abrs00^1\circ\KPt2
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta:\xi\in D\KP1\} }. Noting that
by our assumption for every fixed \math{\xi\in D} we have
that
\math{
(\kern0.37mm\Abrs00^1\circ\KPt2
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\,;\kern0.07mm\mu\,,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)
} is
measurable, the assertion immediately follows.
\end{proof} | 3,022 | 362,273 | en |
train | 0.47.24 | \begin{lemma}\label{Le Nu_1 = sup ...}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ be reflexive with $\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a compatible norm and $\,S$ a separable linear subspace in
$\,\vPi\dlbetss01\,$. Also let \vskip.5mm\centerline{$
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1 = \seqss44{ \sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44
\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I) : u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)} \KP1 $.} \inskipline{.5}0
Then there is a countable $\,D\subseteq\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb10 I$ with
$\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u=
\sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)$ for $\,u\in S\,$.
\end{lemma}
\begin{proof} Putting \math{E=\vPi\dlbetss01\kern0.37mm_{/\,S} } let \math{A} be
countable and \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,dense, and let \math{\scrmt R} be the set
of all pairs \math{(\kern0.37mm u\kern0.37mm,\kern0.07mm\xi\kern0.37mm)\in A\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} with
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss11\xi\le 1} and \mathss35{u\fvalss01\xi=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u
}. By reflexivity and Hahn\,--\,Banach we then have \math{A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt R}
and hence by {\kern0.15mm countable choice\kern0.15mm} there is a function \math{\scrmt P\subseteq
\scrmt R} with \mathss31{A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt P}. Let \mathss31{D={}^{}{\rm rng}\,{}_{{}^{}}\scrmt P}.
Now for all \math{u\in S} we trivially have \mathss38{\sup \KPt8(\kern0.37mm
\Abrs00^1\circ\kern0.15mm u\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)\le\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u }, \,and hence
assuming that \math{\sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)
< \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u } holds for some \mathss34{u\in S}, \,it suffices to
get a contradiction. Taking \math{ \varepsilon = \frac 12\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}
u - \sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)) } we first find
some \math{v\in A} with \mathss34{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm u - v\kern0.37mm) < \varepsilon}.
Then for \math{\xi=\scrmt P\fvalss30 v} we have \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss11\xi\le 1}
and \mathss34{v\fvalss01\xi=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} v}, \,and hence \ $
\sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)
= \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u - 2\KPt8\varepsilon
< \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u - \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm u - v\kern0.37mm) - \varepsilon$ \inskipline{.25}{35.7}
${}\le\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} v - \varepsilon = v\fvalss01\xi - \varepsilon =
(\kern0.37mm v - u\kern0.37mm)\fvalss01\xi + u\fvalss01\xi - \varepsilon $ \inskipline{.25}{35.7}
${}\le|\KP1 u\fvalss01\xi\KP1| + \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm u - v\kern0.37mm) -\varepsilon
< |\KP1 u\fvalss01\xi\KP1| \KP1 $, \,a contradiction.
\end{proof} | 2,876 | 362,273 | en |
train | 0.47.25 | \insubsubhead Decomposable positive measures \label{Ss decos}
Decomposability, as well as being {\sl almost decomposable\kern0.15mm}, is a property
for a positive measure $\mu$ that is weaker than the usual
\q{$\sigma\,$-\,finiteness} which we call \rsigma6finiteness, and that is
sufficiently strong still to have \mathss38{\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.15mm) }
canonically represent the strong dual of \mathss38{\mLrs42^1(\kern0.37mm\mu\kern0.15mm) }.
For example Haar measures of suitably \q{large} locally compact topological
groups are almost decomposable but not \rsigma6finite. See
Example \ref{Exa Haar} on page \pageref{Exa Haar} below for some details
concerning this assertion.
\begin{definitions}\label{df decomp}
(1) \ Say that \math{N\kern0.15mmrim1} is \mathss37{\mu}{\it--\,negligible\kern0.37mm} if{}f \math{
\mu} is a positive measure with \inskipline09
$N\kern0.15mmrim1\subseteq \kern0.15mm\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ and \mathss38{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+\kern-0.63mm\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}\kern0.15mm N\kern0.15mmrim1 \subseteq \kern0.37mm
\bigcup\KPt8\{\KPt8\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N\kern-0.3mm:N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} }, \inskipline{.5}2
(2) \ For a positive measure \math{\mu} on \math{{}^{}\Cal Omega} say that \math{\mu}
is {\it almost decomposable\kern0.37mm} if{}f there are \mathss30{\scrmt A \subseteq \label{decos A}
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+} and \math{N\kern0.15mmrim1} with \math{
\scrmt A\cupss42\{\kern0.37mm N\kern0.15mmrim1\kern0.15mm\} } disjoint and \math{ {}^{}\Cal Omega =
\bigcup\,\scrmt A\cupss42 N\kern0.15mmrim1 } and such that \math{N\kern0.15mmrim1} is \mathss37{
\mu}--\,negligible, and such that also \math{N\kern0.15mmrimm1} is \mathss37{\mu
}--\,negligible whenever \ú$\kern0.37mm N\kern0.15mmrimm1\subseteq{}^{}\Cal Omega$ \linebreak
is such that for every \math{
A\in\scrmt A} there is \math{N} with \mathss38{A \capss31 N\kern0.15mmrimm1 \subseteq N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \inskipline{.5}2
(3) \ For a positive measure \math{\mu} on \math{{}^{}\Cal Omega} say that \math{\mu}
is {\it decomposable\kern0.37mm} if{}f there is some disjoint \math{\scrmt A \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+} with \math{{}^{}\Cal Omega=\bigcup\,\scrmt A } and such
that every \math{N\kern0.15mmrim1\subseteq{}^{}\Cal Omega} is \mathss37{\mu}--\,negligible whenever \math{
\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 N\kern0.15mmrim1 \subseteq \kern0.37mm \bigcup\KPt8\{\KPt8\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N \kern-0.3mm :
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} } holds, \inskipline{.5}2
(4) \ For a positive measure \math{\mu} on \math{{}^{}\Cal Omega} say that \math{\mu}
is {\it truly decomposable\kern0.37mm} if{}f there is some disjoint \math{
\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+} with \math{{}^{}\Cal Omega=\bigcup\,\scrmt A }
and such that \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } holds for every
\math{N\subseteq{}^{}\Cal Omega} with \mathss38{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 N \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
\end{definitions}
Trivially \rsigma6finite positive measures are truly decomposable, and these
in turn are decomposable by
Proposition \ref{Propo top-deco} below. If
\math{\mu} is a positive measure on \math{{}^{}\Cal Omega} such that \mathss03{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+=\emptyset} holds, \,then trivially every \math{
A\subseteq{}^{}\Cal Omega} is \mathss37{\mu}--\,negligible, and hence \math{\mu} is
almost decomposable. A positive measure \math{\mu} on \math{ {}^{}\Cal Omega =
\mathbb R\kern-.2mm\times\kern-.2mm\mathbb R} that is decomposable but not truly decomposable is given in
Example \ref{Exa not trul deco} on page \pageref{Exa not trul deco} below. It
seems to be quite difficult to find positive measures that {\sl are not\kern0.15mm}
almost decomposable.
See Problem \ref{Prblm z-z mea} on page \pageref{Prblm z-z mea}
as well as the subsequent examples and problems.
\begin{definitions}\label{df top deco}
(1) \ Say that \math{\scrmt T} {\it positively almost \eit Radonizes\kern0.37mm} \math{
\mu} if{}f there is \math{{}^{}\Cal Omega} with \math{\mu} a \linebreak
positive measure on \math{
{}^{}\Cal Omega} and \math{(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\scrmt T\,) } a locally compact Hausdorff
topological space such that for \math{\scrmt K=\{\,K:K\kern0.37mm\text{ is \mathss37{
\scrmt T}--\,compact } \} } it holds that \math{ \scrmt K \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } and also for all \math{ A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } it holds that \mathss38{ \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A =
\sup\KPt8\{\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K:K\in\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\KP1\} }, \inskipline{.5}2
(2) \ Say that \math{\scrmt T} {\it positively \eit Radonizes\kern0.37mm} \math{\mu \label{df pos rdz} }
if{}f there is \math{{}^{}\Cal Omega} with \math{\mu} a positive measure on \math{
{}^{}\Cal Omega} and \math{(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\scrmt T\,) } a locally compact Hausdorff
topological space such that for \ú$\kern0.37mm\scrmt K={\kern-0.63mm}$ \linebreak
\ú$\{\,K:K\kern0.37mm\text{ is \mathss37{
\scrmt T}--\,compact } \} \kern0.37mm$ it holds that \math{\scrmt T\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } and \math{
\scrmt K\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } and also for all \math{ U \kern-0.3mm\in
\scrmt T} it holds that \mathss38{\mu\fvalss01 U = \sup\KPt8\{\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K:
K\in\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm U\KP1\} } and for all \ú$\kern0.37mm A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern-0.63mm$ \linebreak
it holds that \mathss38{ \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A
= \inf\,\{\,\mu\fvalss01 U:A\subseteq U\in\scrmt T\KP1\} }, \inskipline{.5}2
(3) \ Say that \math{\mu} is {\it positive almost \eit Radonian\kern0.37mm} if{}f \inskipline0{11}
there is \math{\scrmt T} such that \math{\scrmt T} positively almost
\erm Radonizes \mathss35{\mu}, \inskipline{.5}2
(4) \ Say that \math{\mu} is {\it positive \eit Radonian\kern0.37mm} if{}f \label{df pos Radon} \inskipline0{11}
there is \math{\scrmt T} such that \math{\scrmt T} positively
\erm Radonizes \mathss35{\mu}, \inskipline{.5}2 | 3,507 | 362,273 | en |
train | 0.47.26 | \begin{definitions}\label{df top deco}
(1) \ Say that \math{\scrmt T} {\it positively almost \eit Radonizes\kern0.37mm} \math{
\mu} if{}f there is \math{{}^{}\Cal Omega} with \math{\mu} a \linebreak
positive measure on \math{
{}^{}\Cal Omega} and \math{(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\scrmt T\,) } a locally compact Hausdorff
topological space such that for \math{\scrmt K=\{\,K:K\kern0.37mm\text{ is \mathss37{
\scrmt T}--\,compact } \} } it holds that \math{ \scrmt K \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } and also for all \math{ A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } it holds that \mathss38{ \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A =
\sup\KPt8\{\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K:K\in\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\KP1\} }, \inskipline{.5}2
(2) \ Say that \math{\scrmt T} {\it positively \eit Radonizes\kern0.37mm} \math{\mu \label{df pos rdz} }
if{}f there is \math{{}^{}\Cal Omega} with \math{\mu} a positive measure on \math{
{}^{}\Cal Omega} and \math{(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\scrmt T\,) } a locally compact Hausdorff
topological space such that for \ú$\kern0.37mm\scrmt K={\kern-0.63mm}$ \linebreak
\ú$\{\,K:K\kern0.37mm\text{ is \mathss37{
\scrmt T}--\,compact } \} \kern0.37mm$ it holds that \math{\scrmt T\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } and \math{
\scrmt K\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } and also for all \math{ U \kern-0.3mm\in
\scrmt T} it holds that \mathss38{\mu\fvalss01 U = \sup\KPt8\{\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K:
K\in\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm U\KP1\} } and for all \ú$\kern0.37mm A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern-0.63mm$ \linebreak
it holds that \mathss38{ \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A
= \inf\,\{\,\mu\fvalss01 U:A\subseteq U\in\scrmt T\KP1\} }, \inskipline{.5}2
(3) \ Say that \math{\mu} is {\it positive almost \eit Radonian\kern0.37mm} if{}f \inskipline0{11}
there is \math{\scrmt T} such that \math{\scrmt T} positively almost
\erm Radonizes \mathss35{\mu}, \inskipline{.5}2
(4) \ Say that \math{\mu} is {\it positive \eit Radonian\kern0.37mm} if{}f \label{df pos Radon} \inskipline0{11}
there is \math{\scrmt T} such that \math{\scrmt T} positively
\erm Radonizes \mathss35{\mu}, \inskipline{.5}2
(5) \ Say that \math{\mu} is {\it topologically almost decomposable\kern0.37mm} if{}f
there are \math{\scrmt A\,,\kern0.15mm\scrmt T\kern0.07mm,\kern0.15mm N\kern0.15mmrim1} such that \math{
\scrmt T} positively almost \erm Radonizes \mathss35{\mu} and for \math{
\scrmt K = \{\,K:K\kern0.37mm\text{ is \mathss37{\scrmt T}--\,compact } \} } it holds
that \math{\scrmt A\subseteq\scrmt K\kern0.15mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } and \math{
\scrmt K\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 N\kern0.15mmrim1 \subseteq \kern0.37mm \bigcup\KPt8\{\, \Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N : N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} } and \math{
\scrmt A\cupss42\{\,N\kern0.15mmrim1\kern0.15mm\} } is disjoint with \math{\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu =
\bigcup\,\scrmt A\cupss42 N\kern0.15mmrim1 } and \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K} is
countable for all \math{K\in\scrmt K} and also \math{ A\capss33 U \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } holds for all \math{
A\in\scrmt A} and \mathss30{U\kern-0.3mm\in\scrmt T}, \inskipline{.5}2
(6) \ Say that \math{\mu} is {\it topologically decomposable\kern0.37mm} if{}f there
are \math{\scrmt A\,,\kern0.15mm\scrmt T\kern0.07mm,\kern0.15mm N\kern0.15mmrim1} such that \math{
\scrmt T} positively \erm Radonizes \mathss35{\mu} and such that for
\math{\scrmt K=
\{\,K:K\kern0.37mm\text{ is \mathss37{\scrmt T}--\,compact } \} } it
holds that $\scrmt A\subseteq\scrmt K\kern0.15mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}$ and
$\scrmt K\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 N\kern0.15mmrim1\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ and
$\scrmt A\cupss42\{\,N\kern0.15mmrim1\kern0.15mm\}$ is disjoint with
$\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu=\bigcup\,\scrmt A\cupss42 N\kern0.15mmrim1$
and \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K} is countable for all \math{K\in\scrmt K}
and \math{
A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } holds for
all \math{A\in\scrmt A} and \mathss30{U\kern-0.3mm\in\scrmt T}.
\end{definitions}
Note that the condition \math{ A\capss33 U \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } in (5) and (6) of
Definitions \ref{df top deco} above means that we have \math{ A\capss33 U \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} or \math{A\capss33 U \in 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} =
\{\kern0.37mm\emptyset\kern0.37mm\} } which in turn is equivalent to having \math{
0 < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm) < \lower1.05mm\hbox{$^+$}\infty} or \mathss35{ A\capss33 U =
\emptyset}. Note also the impli- cations \mathss35{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm) \in \rbb R^+ \impss13
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm) \not= \hbox{\font\SweD =cmssbx10\SweD U}{} \impss33
A\capss33 U\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}. That positive (\kern0.15mm almost\kern0.15mm) \erm Radonian measures
are topologically (\kern0.15mm almost\kern0.15mm) decomposable, and that these in turn are
almost decomposable is seen from the next
\begin{proposition}\label{Propo top-deco}
For the properties given below the implications \ú$\,(5)\impss11(6)\impss11(7)$
and $\,(3)\impss11(4)\impss11(7)$ and $\,(1)\impss11[\KPp1.4(2)\text{ and }(3)\KPp1.4
]$ and $\,(2)\impss11(4)$ hold. {\rm\inskipline{.6}4
(1) \ }$\mu$ is positive \eit Radonian{\kern0.15mm\rm,\inskipline{.2}4
(2) \ }$\mu$ is positive almost \eit Radonian{\kern0.15mm\rm,\inskipline{.2}4
(3) \ }$\mu$ is topologically decomposable{\kern0.15mm\rm,\inskipline{.2}4
(4) \ }$\mu$ is topologically almost decomposable{\kern0.15mm\rm,\inskipline{.2}4
(5) \ }$\mu$ is truly decomposable{\kern0.15mm\rm,\inskipline{.2}4
(6) \ }$\mu$ is decomposable{\kern0.15mm\rm,\inskipline{.2}4
(7) \ }$\mu$ is almost decomposable.
\end{proposition} | 3,706 | 362,273 | en |
train | 0.47.27 | $\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu=\bigcup\,\scrmt A\cupss42 N\kern0.15mmrim1$
and \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K} is countable for all \math{K\in\scrmt K}
and \math{
A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } holds for
all \math{A\in\scrmt A} and \mathss30{U\kern-0.3mm\in\scrmt T}.
\end{definitions}
Note that the condition \math{ A\capss33 U \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } in (5) and (6) of
Definitions \ref{df top deco} above means that we have \math{ A\capss33 U \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} or \math{A\capss33 U \in 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} =
\{\kern0.37mm\emptyset\kern0.37mm\} } which in turn is equivalent to having \math{
0 < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm) < \lower1.05mm\hbox{$^+$}\infty} or \mathss35{ A\capss33 U =
\emptyset}. Note also the impli- cations \mathss35{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm) \in \rbb R^+ \impss13
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm) \not= \hbox{\font\SweD =cmssbx10\SweD U}{} \impss33
A\capss33 U\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}. That positive (\kern0.15mm almost\kern0.15mm) \erm Radonian measures
are topologically (\kern0.15mm almost\kern0.15mm) decomposable, and that these in turn are
almost decomposable is seen from the next
\begin{proposition}\label{Propo top-deco}
For the properties given below the implications \ú$\,(5)\impss11(6)\impss11(7)$
and $\,(3)\impss11(4)\impss11(7)$ and $\,(1)\impss11[\KPp1.4(2)\text{ and }(3)\KPp1.4
]$ and $\,(2)\impss11(4)$ hold. {\rm\inskipline{.6}4
(1) \ }$\mu$ is positive \eit Radonian{\kern0.15mm\rm,\inskipline{.2}4
(2) \ }$\mu$ is positive almost \eit Radonian{\kern0.15mm\rm,\inskipline{.2}4
(3) \ }$\mu$ is topologically decomposable{\kern0.15mm\rm,\inskipline{.2}4
(4) \ }$\mu$ is topologically almost decomposable{\kern0.15mm\rm,\inskipline{.2}4
(5) \ }$\mu$ is truly decomposable{\kern0.15mm\rm,\inskipline{.2}4
(6) \ }$\mu$ is decomposable{\kern0.15mm\rm,\inskipline{.2}4
(7) \ }$\mu$ is almost decomposable.
\end{proposition}
\begin{proof} For \math{(1)\impss11(2)} letting \math{\scrmt K\,,\kern0.07mm\scrmt T\kern0.15mm
,\kern0.15mm\mu} be as in Definitions \ref{df top deco}\,(2) above, we need to verify
that for \math{ A \in \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } we have \mathss38{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A = \sup\KPt8\{\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K:K\in\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\KP1\}
}. Thus for given \math{\varepsilon\in\rbb R^+} it suffices to find some \math{K\aR 1
\in\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A} with \mathss34{\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A - \varepsilon <
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K\aR 1}. Now, we first find some set \math{U\kern-0.3mm\in\scrmt T} with \math{
A\subseteq U} and \mathss35{\mu\fvalss01 U < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A + \frac 13\KP1\varepsilon}.
Then we find some \math{V\kern-0.3mm\in\scrmt T} with \math{U\kern0.37mm\setminus A\subseteq V}
and \mathss35{\mu\fvalss01 V <
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm U\kern0.37mm\setminus A\kern0.37mm) + \frac 13\KP1\varepsilon}. We further find
some \math{K\in\scrmt K} with \math{K\subseteq U} and \mathss31{
\mu\fvalss01 U - \frac 13\KP1\varepsilon < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K}, \,and taking \math{K\aR 1 =
K\kern0.37mm\setminus V} we now see that \math{K\aR 1\in\scrmt K} holds with \mathss36{
K\aR 1\subseteq A}. Furthermore, we have \inskipline{.5}{14.6}
$\mu\fvalss01 V <
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm U\kern0.37mm\setminus A\kern0.37mm) + \frac 13\KP1\varepsilon
=\mu\fvalss01 U - \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A + \frac 13\KP1\varepsilon < \frac 23\KP1\varepsilon
$ \ and hence \vskip.2mm\centerline{$
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A - \varepsilon \le \mu\fvalss01 U - \varepsilon
= \mu\fvalss01 U - \frac 13\KP1\varepsilon - \frac 23\KP1\varepsilon
< \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K - \mu\fvalss01 V \le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K\aR 1\,$.} \vskip.5mm
Having the above, for the proofs of \math{(1)\impss11(3)} and \math{
(2)\impss11(4)} letting \math{\scrmt K\,,\kern0.07mm\scrmt T\kern0.15mm,\kern0.15mm\mu} be as in
Definitions \ref{df top deco}\,(5) above, it suffices to show existence of \math{
\scrmt A} and \math{N\kern0.15mmrim1} such that we have \math{ \scrmt A \subseteq
\scrmt K\kern0.15mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } and \math{\scrmt K\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 N\kern0.15mmrim1 \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and such that \math{
\scrmt A\cupss42\{\,N\kern0.15mmrim1\kern0.15mm\} } is disjoint with \math{ \bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu =
\bigcup\,\scrmt A\cupss42 N\kern0.15mmrim1 } and for all \math{K\in\scrmt K} we have
that \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K} is countable and also \math{
A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } holds for
all \math{A\in\scrmt A} and \mathss30{U\kern-0.3mm\in\scrmt T}.
To get such \math{\scrmt A\,,\kern0.15mm N\kern0.15mmrim1} we let \math{\Cal K} be the set of
all disjoint \math{\scrmt A\subseteq\scrmt K\kern0.15mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } with the
property that \math{ A\capss33 U \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } holds for all \math{A\in
\scrmt A} and \mathss30{U\kern-0.3mm\in\scrmt T}. Since trivi- ally \math{\emptyset
\in\Cal K} holds, by {\sl Zorn's lemma\kern0.15mm} there is some \math{\scrmt A} that
is maximal in \mathss34{\Cal K}. Then we take \mathss36{ N\kern0.15mmrim1 =
\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.07mm\setminus\kern0.15mm\bigcup\,\scrmt A }.
We first show that \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K} is countable for all \mathss35{
K\in\scrmt K}. So we fix \math{K} and using local compactness of \math{
\scrmt T} find a relatively \mathss37{\scrmt T}--\,compact \math{ U \kern-0.3mm \in
\scrmt T} with \mathss30{K\subseteq U}. Then we have \math{
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm)
\le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} U < \lower1.05mm\hbox{$^+$}\infty } from which it follows that the set \vskip.3mm
$\scrmt A\capss31\{\,A:\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm)
\not=0\KP1\} \kern0.37mm$ is countable. Since we have | 4,025 | 362,273 | en |
train | 0.47.28 | Having the above, for the proofs of \math{(1)\impss11(3)} and \math{
(2)\impss11(4)} letting \math{\scrmt K\,,\kern0.07mm\scrmt T\kern0.15mm,\kern0.15mm\mu} be as in
Definitions \ref{df top deco}\,(5) above, it suffices to show existence of \math{
\scrmt A} and \math{N\kern0.15mmrim1} such that we have \math{ \scrmt A \subseteq
\scrmt K\kern0.15mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } and \math{\scrmt K\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 N\kern0.15mmrim1 \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and such that \math{
\scrmt A\cupss42\{\,N\kern0.15mmrim1\kern0.15mm\} } is disjoint with \math{ \bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu =
\bigcup\,\scrmt A\cupss42 N\kern0.15mmrim1 } and for all \math{K\in\scrmt K} we have
that \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K} is countable and also \math{
A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } holds for
all \math{A\in\scrmt A} and \mathss30{U\kern-0.3mm\in\scrmt T}.
To get such \math{\scrmt A\,,\kern0.15mm N\kern0.15mmrim1} we let \math{\Cal K} be the set of
all disjoint \math{\scrmt A\subseteq\scrmt K\kern0.15mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } with the
property that \math{ A\capss33 U \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } holds for all \math{A\in
\scrmt A} and \mathss30{U\kern-0.3mm\in\scrmt T}. Since trivi- ally \math{\emptyset
\in\Cal K} holds, by {\sl Zorn's lemma\kern0.15mm} there is some \math{\scrmt A} that
is maximal in \mathss34{\Cal K}. Then we take \mathss36{ N\kern0.15mmrim1 =
\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.07mm\setminus\kern0.15mm\bigcup\,\scrmt A }.
We first show that \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K} is countable for all \mathss35{
K\in\scrmt K}. So we fix \math{K} and using local compactness of \math{
\scrmt T} find a relatively \mathss37{\scrmt T}--\,compact \math{ U \kern-0.3mm \in
\scrmt T} with \mathss30{K\subseteq U}. Then we have \math{
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm)
\le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} U < \lower1.05mm\hbox{$^+$}\infty } from which it follows that the set \vskip.3mm
$\scrmt A\capss31\{\,A:\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm)
\not=0\KP1\} \kern0.37mm$ is countable. Since we have
an injection $\kern0.37mm \scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K\kern0.37mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}
\to
\scrmt A\capss31\{\,A:\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm)
\not=0\KP1\} $
\noindent
given by \mathss35{A\capss31 K\mapsto A}, \,the assertion follows.
To establish \math{ \scrmt K\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 N\kern0.15mmrim1 \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } indirectly, suppose that we have some \math{
K\aar 0\in\scrmt K} with \mathss38{K\aar 0\capss02 N\kern0.15mmrim1 \not \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }. Since \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K\aar 0}
is countable, we first see
that \math{K\aar 0\capss02 N\kern0.15mmrim1\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} holds, and hence we have
\mathss30{K\aar 0\capss02 N\kern0.15mmrim1\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}. Then
by \math{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm K\aar 0\capss02 N\kern0.15mmrim1\kern0.15mm) =
\sup\KPt8\{\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K:K\in\scrmt K\capss22
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.15mm K\aar 0\capss02 N\kern0.15mmrim1\kern0.15mm)\,\} } we find
some \math{K\aR 1\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm\scrmt K} with
\mathss30{K\aR 1\subseteq K\aar 0\capss02 N\kern0.15mmrim1}, \,and we take
$A = K\aR 1\kern-0.63mm\setminus\kern0.15mm\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1)) $
having now $A\in\scrmt K$ with
$A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}$
for all $U\kern-0.3mm\in\scrmt T$.
Indeed, with \math{U\kern-0.3mm\in\scrmt T} supposing that \math{ \emptyset \not =
A\capss33 U\not\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ } holds, we then have
$A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ , and
{\sl for the moment supposing\kern0.15mm} (\kern0.15mm$*$\kern0.15mm) that also
$K\aR 1\kern-0.63mm\setminus A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$
holds, we
obtain \math{\emptyset\not=A\capss33 U\subseteq
K\aR 1\kern-0.3mm\cap\kern0.37mm U\subseteq\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1))
} and
hence \mathss36{A\capss34\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1))
\not=\emptyset }, \,a {\sl contradiction\kern0.15mm}.
Since now \math{A\capss35\bigcup\,\scrmt A\subseteq K\aR 1\capss05\bigcup\,\scrmt A
\subseteq N\kern0.15mmrim1\capss05\bigcup\,\scrmt A = \emptyset} holds, by the maximality
of \linebreak
$\scrmt A\kern0.37mm$ we have \math{A=\emptyset} and hence \mathss38{ K\aR 1 =
\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1)) }.
From this we see existence of some set \math{V} with \math{K\aR 1\subseteq \kern0.15mm
V\kern-0.3mm\in\scrmt T} and \mathss36{ \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K\aR 1 =
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm V\capss01 K\aR 1)=0 }. Hence we obtain \mathss38{ K\aR 1 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\sbig)0 }, \,a {\sl contradiction\kern0.15mm}. | 4,069 | 362,273 | en |
train | 0.47.29 | having now $A\in\scrmt K$ with
$A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}$
for all $U\kern-0.3mm\in\scrmt T$.
Indeed, with \math{U\kern-0.3mm\in\scrmt T} supposing that \math{ \emptyset \not =
A\capss33 U\not\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ } holds, we then have
$A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ , and
{\sl for the moment supposing\kern0.15mm} (\kern0.15mm$*$\kern0.15mm) that also
$K\aR 1\kern-0.63mm\setminus A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$
holds, we
obtain \math{\emptyset\not=A\capss33 U\subseteq
K\aR 1\kern-0.3mm\cap\kern0.37mm U\subseteq\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1))
} and
hence \mathss36{A\capss34\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1))
\not=\emptyset }, \,a {\sl contradiction\kern0.15mm}.
Since now \math{A\capss35\bigcup\,\scrmt A\subseteq K\aR 1\capss05\bigcup\,\scrmt A
\subseteq N\kern0.15mmrim1\capss05\bigcup\,\scrmt A = \emptyset} holds, by the maximality
of \linebreak
$\scrmt A\kern0.37mm$ we have \math{A=\emptyset} and hence \mathss38{ K\aR 1 =
\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1)) }.
From this we see existence of some set \math{V} with \math{K\aR 1\subseteq \kern0.15mm
V\kern-0.3mm\in\scrmt T} and \mathss36{ \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K\aR 1 =
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm V\capss01 K\aR 1)=0 }. Hence we obtain \mathss38{ K\aR 1 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\sbig)0 }, \,a {\sl contradiction\kern0.15mm}.
So, to finish the indirect proof of
\mathss39{\scrmt K\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 N\kern0.15mmrim1\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}
}, \,we must show that (\kern0.15mm$*$\kern0.15mm) above holds. Indeed,
in the contrary case we have \math{
K\aR 1\kern-0.63mm\setminus A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}
and then, as above, we find
some \math{K\aar 2\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm\scrmt K}
with
\mathss38{K\aar 2\subseteq
K\aR 1\kern-0.63mm\setminus A\capss33 U
\subseteq K\aR 1\kern-0.63mm\setminus A\subseteq
\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1))
}, \,and further
some
\math{V\kern-0.3mm\in\scrmt T} with
\math{K\aar 2\subseteq V\capss01 K\aR 1} and
\mathss36{\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm V\capss01 K\aR 1)=0}.
Hence we obtain
\mathss38{K\aar 2\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \,a
{\sl contradiction\kern0.15mm}.
For \math{(5)\impss11(6)} letting \math{\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm N\kern0.15mmrim1} be as
in Definitions \ref{df decomp}\,(3) above and letting $\kern0.37mm\scrmt A$ \linebreak
be as in
(4) there, using the {\sl axiom of choice\kern0.15mm} we find \math{N} with \math{
N\kern0.15mmrim1\subseteq N\subseteq{}^{}\Cal Omega } and such that \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 N \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } holds. Now it easily follows that \math{
N\kern0.15mmrim1} is \mathss37{\mu}--\,negligible.
For \math{(6)\impss11(7)} letting \math{\mu\,,\kern0.15mm{}^{}\Cal Omega} be as
in Definitions \ref{df decomp}\,(2) and letting \math{\scrmt A\kern0.15mmrim0} stand
for the \math{\scrmt A} in (3) there, we take \math{ \scrmt A =
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+\kern-0.2mm\cap\KPt5\scrmt A\kern0.15mmrim0 } and \mathss38{
N\kern0.15mmrim1=\kern0.37mm\bigcup\KP1(\kern0.37mm\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22
\scrmt A\kern0.15mmrim0\kern0.15mm) }. It is now a simple exercise to show that \math{
N\kern0.15mmrim1} is \mathss37{\mu}--\,negligible, and that also the condition
concerning \math{N\kern0.15mmrimm1} there holds.
By \math{(1)\impss11(2)} we trivially have \mathss37{(3)\impss11(4)}, \,and
for \math{(4)\impss11(7)} letting \ú$\kern0.37mm\scrmt A\,,\kern0.15mm\scrmt K\,,\kern0.15mm\scrmt T\kern0.07mm
,$ \linebreak
\ú$N\kern0.15mmrim1,\kern0.15mm\mu\kern0.37mm$ be as in Definitions \ref{df top deco}\,(5) it
suffices to show that \math{\scrmt A \subseteq \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+ } holds
and that \math{N\kern0.15mmrim1} is \mathss37{
\mu}--\,negligible, and that also \math{N\kern0.15mmrimm1} is \mathss37{\mu
}--\,negligible whenever \mathss30{N\kern0.15mmrimm1\subseteq{}^{}\Cal Omega \kern-0.3mm} \mathss03{
\kern-0.3mm=\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } is
such that for \math{
A\kern0.07mm\ar 1\in\scrmt A} there is \math{N} with \mathss38{
A\kern0.07mm\ar 1\kern-0.2mm\cap\kern0.15mm N\kern0.15mmrimm1 \subseteq N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }. | 3,459 | 362,273 | en |
train | 0.47.30 | For \math{(5)\impss11(6)} letting \math{\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm N\kern0.15mmrim1} be as
in Definitions \ref{df decomp}\,(3) above and letting $\kern0.37mm\scrmt A$ \linebreak
be as in
(4) there, using the {\sl axiom of choice\kern0.15mm} we find \math{N} with \math{
N\kern0.15mmrim1\subseteq N\subseteq{}^{}\Cal Omega } and such that \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 N \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } holds. Now it easily follows that \math{
N\kern0.15mmrim1} is \mathss37{\mu}--\,negligible.
For \math{(6)\impss11(7)} letting \math{\mu\,,\kern0.15mm{}^{}\Cal Omega} be as
in Definitions \ref{df decomp}\,(2) and letting \math{\scrmt A\kern0.15mmrim0} stand
for the \math{\scrmt A} in (3) there, we take \math{ \scrmt A =
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+\kern-0.2mm\cap\KPt5\scrmt A\kern0.15mmrim0 } and \mathss38{
N\kern0.15mmrim1=\kern0.37mm\bigcup\KP1(\kern0.37mm\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22
\scrmt A\kern0.15mmrim0\kern0.15mm) }. It is now a simple exercise to show that \math{
N\kern0.15mmrim1} is \mathss37{\mu}--\,negligible, and that also the condition
concerning \math{N\kern0.15mmrimm1} there holds.
By \math{(1)\impss11(2)} we trivially have \mathss37{(3)\impss11(4)}, \,and
for \math{(4)\impss11(7)} letting \ú$\kern0.37mm\scrmt A\,,\kern0.15mm\scrmt K\,,\kern0.15mm\scrmt T\kern0.07mm
,$ \linebreak
\ú$N\kern0.15mmrim1,\kern0.15mm\mu\kern0.37mm$ be as in Definitions \ref{df top deco}\,(5) it
suffices to show that \math{\scrmt A \subseteq \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+ } holds
and that \math{N\kern0.15mmrim1} is \mathss37{
\mu}--\,negligible, and that also \math{N\kern0.15mmrimm1} is \mathss37{\mu
}--\,negligible whenever \mathss30{N\kern0.15mmrimm1\subseteq{}^{}\Cal Omega \kern-0.3mm} \mathss03{
\kern-0.3mm=\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } is
such that for \math{
A\kern0.07mm\ar 1\in\scrmt A} there is \math{N} with \mathss38{
A\kern0.07mm\ar 1\kern-0.2mm\cap\kern0.15mm N\kern0.15mmrimm1 \subseteq N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
Now for \math{\scrmt A \subseteq \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+ } taking \math{ A \in
\scrmt A} we have \math{A\not=\emptyset } and hence \mathss30{ A =
A\capss32{}^{}\Cal Omega\in\kern-0.3mm} \mathss03{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } whence
\mathss30{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ \kern-0.3mm}. To show that \math{N\kern0.15mmrim1}
is \mathss37{\mu}--\,negligible, given any \mathss03{ A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ } we must find some \math{N} with \mathss38{
A\capss31 N\kern0.15mmrim1\subseteq N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }. Using
{\sl countable choice\kern0.15mm}, we first find an increasing \math{ \bmii8 K \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.37mm\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.37mm) } with \mathss36{ \mu\circ\bmii8 K
\to\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A }. Then we find \math{ \bmii8 N \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.37mm\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\sbig)0 } with \math{
\bmii8 K\fvalss51 i\capss42 N\kern0.15mmrim1\subseteq\bmii8 N\fvalss41 i } for all \mathss36{
i\in\mathbb No}. Now it suffices to take \mathss32{ N =
A\kern0.07mm\setminus\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 K\cupss54\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 N }.
To get the assertion concerning \math{N\kern0.15mmrimm1} given \math{A} we first take \math{
\bmii8 K} as above. Then noting that \math{
\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32(\kern0.37mm\bmii8 K\fvalss51 i\kern0.37mm) } is countable for every \math{
i\in\mathbb No } by {\sl countable choice\kern0.15mm} we get \mathss03{
\bmii8 K\aR 1\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.37mm\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.37mm) } with \math{
\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 K\kern0.37mm\setminus N\kern0.15mmrim1\subseteq
\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 K\aR 1 } and such that for every \mathss30{i\in\mathbb No }
there is some \math{A\kern0.07mm\ar 1\in\scrmt A } with \mathss34{
\bmii8 K\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\subseteq A\kern0.07mm\ar 1 }. Then again by {\sl countable
choice\kern0.15mm} we get \mathss03{ \bmii8 N \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.37mm\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\sbig)0 } with \math{
\bmii8 K\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\capss42 N\kern0.15mmrimm1\subseteq\bmii8 N\fvalss41 i } for
all \mathss36{i\in\mathbb No }. Since we already know that \math{N\kern0.15mmrim1} is \mathss37{
\mu}--\,negligible, we find some \math{N\aar 1} with \mathss38{
A\capss31 N\kern0.15mmrim1\subseteq N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }. Now it
suffices to take \mathss32{ N = A\kern0.07mm\setminus\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 K\cupss54
\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 N\cupss52 N\aar 1 } to get some \math{N} such that we
have \mathss38{ A\capss32 N\kern0.15mmrimm1 \subseteq N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
\end{proof} | 3,359 | 362,273 | en |
train | 0.47.31 | The idea for the proofs of \math{(1)\impss11(3)} and \math{(2)\impss11(4)}
above is taken from \cite[Proposition 4.14.9\kern0.37mm, p.\ 229]{Edw}\,. Note that
the logical structure of these proofs of the implication \math{(\kern0.37mm i\kern0.37mm)
\impss33(\,i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern-0.3mm\lower1mm\hbox{$^{^+}$}\kern0.07mm) } is basically the following: \
{\sl Axioms\kern0.15mm} \ $\vdash\KN{1.64}\RHB{.35}-$ \inskipline{.5}5
$[\KPp1.4(\kern0.37mm i\kern0.37mm)\impss33\eexi{\scrmt Z}\,\mfrak P\kern0.37mm$ and \math{[\KPp1.4
\mfrak Q\kern0.37mm\text{ or }\neg\KP1\mfrak Q\KPp1.4]\KP1]} and \inskipline{.3}5
$[\KP1[\KPp1.4\eexi{\scrmt Z}\,\mfrak P\kern0.37mm\text{ and }\kern0.37mm\mfrak Q\KPp1.4]
\impss33(\,i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern-0.3mm\lower1mm\hbox{$^{^+}$}\kern0.07mm)\KPp1.4] \kern0.37mm$ and \math{[\KP1[\KPp1.4
\aall{\scrmt Z}\,\mfrak P\kern0.37mm\text{ and }\kern0.37mm\neg\KP1 \mfrak Q\KPp1.4]\impss33
\mfrak R\kern0.37mm\text{ or }\neg\KP1\mfrak R\KPp1.4]} \inskipline0{26}
and \math{ [\KPp1.4 \mfrak R \impss33 \mfrak S\ar 0\kern0.37mm\text{ and }\kern0.37mm \neg\KP1
\mfrak S\ar 0 \KP1 ] } and \math{ [\KP1 \neg\KP1\mfrak R \impss33 \mfrak R^*\kern0.15mm\text{
or }\kern0.37mm \neg\KP1\mfrak R^* \KP1 ] } \inskipline0{26}
and \math{ [\KPp1.4 \mfrak R^* \impss33 \mfrak S\ar 1\kern0.15mm\text{ and }\kern0.37mm \neg\KP1
\mfrak S\ar 1 \, ] } and \mathss39{ [\KP1 \neg\KP1 \mfrak R^* \impss33
\mfrak S\ar 2\kern0.37mm\text{ and }\kern0.37mm \neg\KP1\mfrak S\ar 2 \KP1 ] }.
\begin{lemma}[schema]\label{Le deco meas}
Let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega${\,\rm, }and with \ú$\,\bosy K
\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\tvsps0(K)$ hold. Also let \ú$\, x \in \kern0.15mm
^{}^{}\Cal Omega\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi$ and let {\,\rm X} stand for any of {\rm\,\q{almost}} or
{\rm\,\q{almost scalarly}} or {\rm\,\q{almost simply}}. Further{\kern0.15mm\rm, }let $\,
\scrmt A\,,\kern0.15mm N\kern0.15mmrim1$ be as in {\,\rm Definitions \ref{df decomp}\,(2)} on
page {\,\rm\pageref{decos A}} above. If \ú$\,
(\kern0.37mm x\KP1|\KP1 A\KPt8;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\kern0.37mm)$ is {\,\rm X}
measurable for all \ú$\,A\in\scrmt A${\,\rm, }then $\,
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ is finitely {\,\rm X} measurable.
\end{lemma}
\begin{proof} Noting that we can write \q{X measurable} in the form \q{almost
Z}, assuming that \math{
(\kern0.37mm x\KP1|\KP1 A\KPt8;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\kern0.37mm) } is X
measurable for every \mathss36{A\in\scrmt A}, \,for given \linebreak
\mathss03{
A\kern0.15mm\ar 0\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} it suffices to find some \math{
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that for \math{ B =
A\kern0.15mm\ar 0\kern-0.63mm\setminus N } and \math{\mu\ar 0=\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B } and \math{
x\ar 0=x\KP1|\KP1 B} it holds that \math{
(\kern0.37mm x\ar 0\,;\kern0.07mm\mu\ar 0\,,\kern0.07mm\vPi\kern0.37mm)} is Z\kern0.37mm. For this letting \math{
\scrmt N} be the set of all pairs \math{(\kern0.15mm A\,,\kern0.07mm N\aar 1)} with \math{
A\in\scrmt A} and \math{N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and
such that for \math{A\kern0.07mm\ar 1=A\kern0.07mm\setminus N\aar 1 } it holds that \math{
(\kern0.37mm x\KP1|\KP1 A\kern0.07mm\ar 1\kern0.37mm;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.07mm\ar 1\kern0.15mm,\kern0.07mm
\vPi\kern0.37mm) } is Z\kern0.37mm, by our assumption \math{\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt N}
holds, and hence by the {\sl axiom of choice\kern0.15mm} there is a function \math{
\scrmt N\kern0.07mm\ar 0\subseteq\scrmt N } with \mathss36{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt N\kern0.07mm\ar 0}. Then taking \math{ \scrmt A\kern0.15mm\ar 0 =
\scrmt A\capss41\{\,A : A\capss31 A\kern0.15mm\ar 0 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\} } we have \math{\scrmt A\kern0.15mm\ar 0}
countable, and for \math{N\kern0.15mmrimm1 = N\kern0.15mmrim1 \cupss04 \bigcup \KP1 (\kern0.37mm
\scrmt N\kern0.07mm\ar 0\kern-0.3mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm\scrmt A\kern0.15mm\ar 0\kern0.07mm) \cupss24 \bigcup\KP1 (\kern0.37mm
\scrmt A\setminus\scrmt A\kern0.15mm\ar 0\kern0.07mm) \capss21 A\kern0.15mm\ar 0} it holds that \math{
N\kern0.15mmrimm1} is \mathss37{\mu}--\,negligible. Since \math{N\kern0.15mmrim1\subseteq A\kern0.15mm\ar 0}
holds, for some \math{N} we have \mathss38{N\kern0.15mmrim1 \subseteq N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \,and taking \math{B} as above, it is
straightforward to verify that \math{
(\kern0.37mm x\ar 0\,;\kern0.07mm\mu\ar 0\,,\kern0.07mm\vPi\kern0.37mm)} is Z\kern0.37mm.
\end{proof} | 2,871 | 362,273 | en |
train | 0.47.32 | \insubsubhead Integration of scalar functions \label{Ss int scal}
Since for arbitrary functions \math{ u \subseteq
{}^{}\Cal Omega\times[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } we need to consider upper
inte- grals \math{\upint u\rmdss11\mu } where \math{\mu} is a positive measure
on \mathss36{{}^{}\Cal Omega}, \,we here shortly give the asso- ciated formal
definitions in order to make things precise. Note that in the definition of
the Lebesgue quasi\kern0.37mm-\kern0.37mm norm \math{\|\,x\,\|\Lnorss33^p_\mu } in
Constructions \ref{Ctr |x|_lL^p}\,(2) on page \pageref{ctr L^p-norm} above we
already implicitly used the concept of upper integral.
\begin{constructions}[\kern0.15mm positive, real and pseudo\kern0.37mm-\kern0.15mm usual integrals]\label{defi re scal int} $\null$ \vskip.5mm
\begin{enumerate}\begin{myLeftskip}{-4}{.6}{.6}
\item \ $\loint u\rmdss20\mu = \uniqset t: \mu \in \kern0.15mm
^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\mu}\KP1[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KP1] \kern0.37mm$ and \newskline{32}
$ u \in\kern0.15mm ^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\eightmath u}\KP1[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KP1] \kern0.37mm$ and $\kern0.37mm
{{}^{}{\rm dom}\,{}_{{}^{}}} u\subseteq\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu \kern0.37mm$ and \newskline{28}
$ t = \sup\kern0.15mm\big\{\kern0.37mm\sum\,\seq{\KPt8
t\kern-.2mm\cdot\kern-.2mm(\kern0.37mm\mu\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm\sigma\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.15mm t\kern0.37mm\}\kern0.15mm))
: \sigma : t\in{}^{}{\rm rng}\,{}_{{}^{}}\sigma\KPt8} : \mu : $ \newskline{25.5}
$\sigma\in\kern0.15mm^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\eightmath\sigmaa}\KPt8\lbb R_+ \kern0.37mm$ and $\kern0.37mm
{{}^{}{\rm dom}\,{}_{{}^{}}}\sigma\subseteq\{\,\eta:\sigma\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\,\}$ \newfline
and $\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\sigma\kern0.37mm$ is finite and $\kern0.37mm \{\,\sigma\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm
t\kern0.37mm\} : t \in \hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \subseteq {{}^{}{\rm dom}\,{}_{{}^{}}}\mu \,\big\} \KP1 $, \KP9
\item \ $\upint u\rmdss20\mu = \uniqset t : \mu \in \kern0.15mm \label{ctr upint}
^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\mu}\KP1[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KP1] \kern0.37mm$ and \newskline{31.9}
$u \in\kern0.15mm ^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\eightmath u}\KP1[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KP1] \kern0.37mm$ and $\kern0.37mm
{{}^{}{\rm dom}\,{}_{{}^{}}} u\subseteq\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu \kern0.37mm$ and \newskline{11.5}
$ t = \inf\kern0.15mm\big\{\kern0.15mm\loint v\rmdss20\mu:\mu:v\in\kern0.15mm
^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\eightmath v}\KP1[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KP1]\kern0.37mm$ and \newfline
$v\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.07mm\barscTbb_R\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu \kern0.37mm $ and $\kern0.37mm
{{}^{}{\rm dom}\,{}_{{}^{}}} u\subseteq\{\KPt8\eta:u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta \KPt9 \}\,\big\} \KP1 $, \KP9
\item \ $\plusint x\rmdss20\mu = \uniqset t : t \label{def +int}
= \loint x\rmdss20\mu = \upint x\rmdss20\mu\,$,
\item \ $\Reint x\rmdss10\mu=\uniqset\smb I\kern-0.3mm\ar 1\kern-0.3mm:x\kern0.37mm$ a function \label{defi Reint}
and $\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}} x\subseteq[\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\lower1.05mm\hbox{$^+$}\infty\,]\kern0.37mm$ and $\,
\aall{\smb I\kern0.15mm,\kern0.07mm\smb J}$ \newskline{25.7}
$\smb I = \plusint\,\seq{\KP1\sup\,\{\,0\,,x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\,\}:\eta=\eta\KP1}
\rmdss20\mu\kern0.37mm$ and \newfline
$\smb J=\plusint\,\seq{\KP1\sup\,\{\,0\,,\kern.2mm\lower1.05mm\hbox{$^-$}(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\,\} :
\eta=\eta\KP1}\rmdss20\mu\impss22\smb I\kern-0.3mm\ar 1=\smb I-\smb J\,$, \KP9
\item \ $\int_{\,A\,}x\rmdss10\mu=\uniqset\smb I:A\subseteq\bigcup\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ \label{defi ps-usual int}
and $\,\eexi{\vPi\kern0.15mm,\kern0.07mm S}\,x \in \kern0.15mm
^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\eightmath x}\kern0.37mm S \kern0.37mm $ and \newskline{23}
$\vPi\kern0.37mm$ is complex pseudo\kern0.37mm-\kern0.15mm usual and $\kern0.37mm\vPi\not=\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm$
and $\,\big[\ \big[\KP{1.5} S = $ \newskline{9.5}
$\mathbb C\cupss31\{\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\lower1.05mm\hbox{$^+$}\infty\,\}\kern0.37mm$ and $\kern0.37mm
\smb I = \Reint\,(\kern0.15mm\kern.1mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD f\kern.1mm}\roman{Re}\kern.9mm x\,|\,A\kern0.15mm)\rmdss10\mu\kern0.15mm + \kern0.37mm\roman i\,
\Reint\,(\kern0.15mm\kern.1mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD f\kern.1mm}\roman{Im}\kern.9mm x\,|\,A\kern0.15mm)\rmdss10\mu $ \newskline{13}
$\in S \KP{1.5}\big]\kern0.37mm$ or $\kern0.37mm\big[\KP{1.5} x \not= \emptyset\kern0.37mm$ and $\kern0.37mm
\smb I \in S = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.37mm$ and $\, \aall \ell\,
\ell\in\Cal L\,(\kern0.07mm\vPi\Reit2\kern0.37mm,\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)$ \newfline
$\impss02 \ell\kern0.37mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb I = \Reint\,
(\kern0.37mm\ell\kern0.15mm\circ\kern0.07mm x\KP1|\KP1 A\kern0.37mm)\rmdss10\mu\KP{1.5}\big]\ \big]\ $.
\end{myLeftskip}\end{enumerate}
\end{constructions}
From Constructions \ref{defi re scal int}\,(1) we get the {\it lower integral\kern0.37mm}
of a\q{positive} valued function \math{u} with respect to a positive measure \mathss35{
\mu}, \,and \ref{defi re scal int}\,(2) and \ref{defi re scal int}\,(3) give
the corresponding {\it upper\kern0.37mm} and {\it positive\kern0.37mm} integral. The
{\it real\kern0.37mm} integral of an extended real valued function w.r.t.\ a positive
measure is given in \ref{defi re scal int}\,(\ref{defi Reint})\,, and item (5)
defines the {\it pseudo\kern0.37mm-\kern0.15mm usual\,} integral. Without delving in the
relevant formal definition given in \cite{Hif} we shortly remark that
pseudo\kern0.37mm-\kern0.15mm usual spaces \math{E} are such structured vector spaces over
some subfield \math{\bold K} of the complex field \math{\raise1.23mm\hbox{\font\SweD =cmr5\SweD f}\kern.1mm\mathbb C} that e.g.\ we
have unambiguously \math{(\kern0.37mm x + y\kern0.37mm)\svs E = x + y } and \math{
(\kern0.37mm t\,x\kern0.37mm)\svs E = t\,y } for all \math{x\kern0.37mm,\kern0.15mm y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E } and \mathss36{
t\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bold K}. If \linebreak
\ú$\vPi\kern0.37mm$ is pseudo\kern0.37mm-\kern0.15mm usual and \math{I} is any
set with \math{I\in\{\,1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.07mm 2\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.15mm\} } or \mathss31{
3\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\subseteq{}^{}{\rm Card}{}^{}\,\kern0.15mm I}, \,then \ú$\kern0.37mm(\kern0.15mm X\kern0.15mm,\kern0.07mm S\kern0.37mm)$ \linebreak
is
pseudo\kern0.37mm-\kern0.15mm usual for any set \math{S} and any vector substructure \math{X}
of \math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.15mm I\kern0.15mm]_{vs} }.
\begin{definitions} | 3,913 | 362,273 | en |
train | 0.47.33 | $\mathbb C\cupss31\{\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\lower1.05mm\hbox{$^+$}\infty\,\}\kern0.37mm$ and $\kern0.37mm
\smb I = \Reint\,(\kern0.15mm\kern.1mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD f\kern.1mm}\roman{Re}\kern.9mm x\,|\,A\kern0.15mm)\rmdss10\mu\kern0.15mm + \kern0.37mm\roman i\,
\Reint\,(\kern0.15mm\kern.1mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD f\kern.1mm}\roman{Im}\kern.9mm x\,|\,A\kern0.15mm)\rmdss10\mu $ \newskline{13}
$\in S \KP{1.5}\big]\kern0.37mm$ or $\kern0.37mm\big[\KP{1.5} x \not= \emptyset\kern0.37mm$ and $\kern0.37mm
\smb I \in S = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.37mm$ and $\, \aall \ell\,
\ell\in\Cal L\,(\kern0.07mm\vPi\Reit2\kern0.37mm,\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)$ \newfline
$\impss02 \ell\kern0.37mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb I = \Reint\,
(\kern0.37mm\ell\kern0.15mm\circ\kern0.07mm x\KP1|\KP1 A\kern0.37mm)\rmdss10\mu\KP{1.5}\big]\ \big]\ $.
\end{myLeftskip}\end{enumerate}
\end{constructions}
From Constructions \ref{defi re scal int}\,(1) we get the {\it lower integral\kern0.37mm}
of a\q{positive} valued function \math{u} with respect to a positive measure \mathss35{
\mu}, \,and \ref{defi re scal int}\,(2) and \ref{defi re scal int}\,(3) give
the corresponding {\it upper\kern0.37mm} and {\it positive\kern0.37mm} integral. The
{\it real\kern0.37mm} integral of an extended real valued function w.r.t.\ a positive
measure is given in \ref{defi re scal int}\,(\ref{defi Reint})\,, and item (5)
defines the {\it pseudo\kern0.37mm-\kern0.15mm usual\,} integral. Without delving in the
relevant formal definition given in \cite{Hif} we shortly remark that
pseudo\kern0.37mm-\kern0.15mm usual spaces \math{E} are such structured vector spaces over
some subfield \math{\bold K} of the complex field \math{\raise1.23mm\hbox{\font\SweD =cmr5\SweD f}\kern.1mm\mathbb C} that e.g.\ we
have unambiguously \math{(\kern0.37mm x + y\kern0.37mm)\svs E = x + y } and \math{
(\kern0.37mm t\,x\kern0.37mm)\svs E = t\,y } for all \math{x\kern0.37mm,\kern0.15mm y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E } and \mathss36{
t\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bold K}. If \linebreak
\ú$\vPi\kern0.37mm$ is pseudo\kern0.37mm-\kern0.15mm usual and \math{I} is any
set with \math{I\in\{\,1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.07mm 2\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.15mm\} } or \mathss31{
3\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\subseteq{}^{}{\rm Card}{}^{}\,\kern0.15mm I}, \,then \ú$\kern0.37mm(\kern0.15mm X\kern0.15mm,\kern0.07mm S\kern0.37mm)$ \linebreak
is
pseudo\kern0.37mm-\kern0.15mm usual for any set \math{S} and any vector substructure \math{X}
of \math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.15mm I\kern0.15mm]_{vs} }.
\begin{definitions}
(1) \ Say that \math{u} is {\it positive \mathss37{\mu}--\,measurable\kern0.37mm}
if{}f there is \math{{}^{}\Cal Omega} such that \math{\mu} is a positive measure on \math{
{}^{}\Cal Omega} and \math{u} is a function with \math{ u \subseteq
{}^{}\Cal Omega\times[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } and such that \math{
u\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KP1 r\kern0.15mm,\lower1.05mm\hbox{$^+$}\infty\KPt9]\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} holds for all \mathss30{
r\in\rbb R^+}, \inskipline{.5}2
(2) \ Say that \math{u} is {\it fully positive \mathss37{\mu}--\,measurable\kern0.37mm}
if{}f \inskipline0{23.6}
$u\kern0.37mm$ is positive \mathss37{\mu}--\,measurable with \mathss36{ {{}^{}{\rm dom}\,{}_{{}^{}}} u = \kern0.15mm
\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}, \inskipline{.5}2
(3) \ Say that \math{\sigma} is {\it positive \mathss37{\mu}--\,simple\kern0.37mm}
if{}f
there is \math{{}^{}\Cal Omega} such that \math{\mu} is a positive measure on \math{
{}^{}\Cal Omega} and \math{\sigma} is a function with \math{\sigma\subseteq{}^{}\Cal Omega\times
\lbb R_+ } and such that \math{{}^{}{\rm rng}\,{}_{{}^{}}\sigma} is finite and also \math{
\{\,\sigma\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm t\kern0.37mm\} : t \in \hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \subseteq {{}^{}{\rm dom}\,{}_{{}^{}}}\mu }
holds.
\end{definitions}
Thus in the case where \math{\mu} is a positive measure, in
Constructions \ref{defi re scal int}\,(\ref{ctr upint}) above we have \math{t}
the infimum of the set of lower integrals of all positive \mathss37{\mu
}--\,measurable functions \math{v} dominating \math{u} in the sence that \math{
u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta} holds for all \mathss34{\eta\in{{}^{}{\rm dom}\,{}_{{}^{}}} u}.
\begin{lemma}\label{Le +int}
Let $\,\mu$ be a positive measure. Then for all $\,x$ the equivalences \vskip.5mm\centerline{$
\plusint x\rmdss20\mu \not= \hbox{\font\SweD =cmssbx10\SweD U}{} \equivss33 0\le\plusint x\rmdss20\mu =
\loint x\rmdss20\mu = \upint x\rmdss20\mu\le\lower1.05mm\hbox{$^+$}\infty $} \inskipline{.5}{20}
and $ \KPp18.6 0\le\plusint x\rmdss20\mu < \lower1.05mm\hbox{$^+$}\infty \equivss33 (\kern0.15mm*\kern0.15mm)
\null
$ hold when $\,(\kern0.15mm*\kern0.15mm)$ \inskipline{.5}0
means that there exist positive $\,\mu\,$--\,measurable functions $\,u\kern0.37mm,\kern0.15mm
v$ with \ú$\,u\le x\le v$ and $\,\loint u\rmdss20\mu \not= \lower1.05mm\hbox{$^+$}\infty$ and $\,
v\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.2mm\setminus\kern-0.3mm{{}^{}{\rm dom}\,{}_{{}^{}}} u\cupss31\{\,\eta :
v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\not=u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} \KP1 $.
\end{lemma}
\begin{proof} Assuming \math{\plusint x\rmdss20\mu \not= \hbox{\font\SweD =cmssbx10\SweD U}{} } we have \math{
\plusint x\rmdss20\mu =\loint x\rmdss20\mu = \upint x\rmdss20\mu = t } for
some \mathss36{t\in\hbox{\font\SweD =cmssbx10\SweD U}{}}. Then taking \math{\sigma = \emptyset } in
Constructions \ref{defi re scal int}\,(1) we see that \math{t = \sup\kern0.37mm A }
for some \math{A} with \math{0\in A\subseteq[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } whence \math{
0\le t\le\lower1.05mm\hbox{$^+$}\infty } follows. Conversely, if we have \mathss35{ 0 \le
\plusint x\rmdss20\mu\le\lower1.05mm\hbox{$^+$}\infty }, \,then \math{ \plusint x\rmdss20\mu \in
\hbox{\font\SweD =cmssbx10\SweD U}{} } and hence \mathss35{\plusint x\rmdss20\mu\not=\hbox{\font\SweD =cmssbx10\SweD U}{} }.
Assuming \math{0\le\plusint x\rmdss20\mu < \lower1.05mm\hbox{$^+$}\infty } from (1) and
(\ref{ctr upint}) in Constructions \ref{defi re scal int} we see existence of
sequences \math{\bosy u} of positive \mathss37{\mu}--\,simple and \math{
\bosy v} of positive \mathss37{\mu}--\,measurable functions with \math{
\lim\sbi{\kern0.15mm i\kern0.37mm\to\kern0.37mm\infty}\kern0.15mm\loint\bosy w\fvalss01 i\rmdss11\mu=0 } for \math{
\bosy w=\seq{\seqss33{
\bosy v\fvalss01 i\fvalss10\eta - \bosy u\fvalss01 i\fvalss10\eta:
\eta=\eta}:i\in\mathbb No\,} } and
\math{\loint\bosy v\fvalss01\emptyset\rmdss11\mu < \lower1.05mm\hbox{$^+$}\infty} and such that
\math{\bosy u\fvalss01 i\le\bosy u\fvalss01 i\kern0.37mm\lower1mm\hbox{$^{^+}$}\le x\le
\bosy v\fvalss01 i\kern0.37mm\lower1mm\hbox{$^{^+}$}\le\bosy v\fvalss01 i} holds for all
\mathss36{i\in\mathbb No}. Taking then
\math{u=\seqss43{\sup\KPt8\{\KPt8\bosy u\fvalss01 i\fvalss10\eta:
i\in\mathbb No\kern0.37mm\}:\bosy u:\eta\in\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bosy u} }
and
\mathss39{v=\langle\KP1\inf\,\{\KPt8\bosy v\fvalss01 i\fvalss10\eta:
i\in\mathbb No\kern0.37mm\}:\bosy v:\eta\in\bigcap\KPt8\{\,{{}^{}{\rm dom}\,{}_{{}^{}}} v:
v\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy v\KPt8\} \,\big\rangle }, \,now
$u$ and $v$ are positive $\mu\,$--\,measurable and hence | 4,004 | 362,273 | en |
train | 0.47.34 | and $ \KPp18.6 0\le\plusint x\rmdss20\mu < \lower1.05mm\hbox{$^+$}\infty \equivss33 (\kern0.15mm*\kern0.15mm)
\null
$ hold when $\,(\kern0.15mm*\kern0.15mm)$ \inskipline{.5}0
means that there exist positive $\,\mu\,$--\,measurable functions $\,u\kern0.37mm,\kern0.15mm
v$ with \ú$\,u\le x\le v$ and $\,\loint u\rmdss20\mu \not= \lower1.05mm\hbox{$^+$}\infty$ and $\,
v\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.2mm\setminus\kern-0.3mm{{}^{}{\rm dom}\,{}_{{}^{}}} u\cupss31\{\,\eta :
v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\not=u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} \KP1 $.
\end{lemma}
\begin{proof} Assuming \math{\plusint x\rmdss20\mu \not= \hbox{\font\SweD =cmssbx10\SweD U}{} } we have \math{
\plusint x\rmdss20\mu =\loint x\rmdss20\mu = \upint x\rmdss20\mu = t } for
some \mathss36{t\in\hbox{\font\SweD =cmssbx10\SweD U}{}}. Then taking \math{\sigma = \emptyset } in
Constructions \ref{defi re scal int}\,(1) we see that \math{t = \sup\kern0.37mm A }
for some \math{A} with \math{0\in A\subseteq[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } whence \math{
0\le t\le\lower1.05mm\hbox{$^+$}\infty } follows. Conversely, if we have \mathss35{ 0 \le
\plusint x\rmdss20\mu\le\lower1.05mm\hbox{$^+$}\infty }, \,then \math{ \plusint x\rmdss20\mu \in
\hbox{\font\SweD =cmssbx10\SweD U}{} } and hence \mathss35{\plusint x\rmdss20\mu\not=\hbox{\font\SweD =cmssbx10\SweD U}{} }.
Assuming \math{0\le\plusint x\rmdss20\mu < \lower1.05mm\hbox{$^+$}\infty } from (1) and
(\ref{ctr upint}) in Constructions \ref{defi re scal int} we see existence of
sequences \math{\bosy u} of positive \mathss37{\mu}--\,simple and \math{
\bosy v} of positive \mathss37{\mu}--\,measurable functions with \math{
\lim\sbi{\kern0.15mm i\kern0.37mm\to\kern0.37mm\infty}\kern0.15mm\loint\bosy w\fvalss01 i\rmdss11\mu=0 } for \math{
\bosy w=\seq{\seqss33{
\bosy v\fvalss01 i\fvalss10\eta - \bosy u\fvalss01 i\fvalss10\eta:
\eta=\eta}:i\in\mathbb No\,} } and
\math{\loint\bosy v\fvalss01\emptyset\rmdss11\mu < \lower1.05mm\hbox{$^+$}\infty} and such that
\math{\bosy u\fvalss01 i\le\bosy u\fvalss01 i\kern0.37mm\lower1mm\hbox{$^{^+}$}\le x\le
\bosy v\fvalss01 i\kern0.37mm\lower1mm\hbox{$^{^+}$}\le\bosy v\fvalss01 i} holds for all
\mathss36{i\in\mathbb No}. Taking then
\math{u=\seqss43{\sup\KPt8\{\KPt8\bosy u\fvalss01 i\fvalss10\eta:
i\in\mathbb No\kern0.37mm\}:\bosy u:\eta\in\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bosy u} }
and
\mathss39{v=\langle\KP1\inf\,\{\KPt8\bosy v\fvalss01 i\fvalss10\eta:
i\in\mathbb No\kern0.37mm\}:\bosy v:\eta\in\bigcap\KPt8\{\,{{}^{}{\rm dom}\,{}_{{}^{}}} v:
v\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy v\KPt8\} \,\big\rangle }, \,now
$u$ and $v$ are positive $\mu\,$--\,measurable and hence
$\{\KPt8
v\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.2mm\setminus\kern-0.3mm{{}^{}{\rm dom}\,{}_{{}^{}}} u\kern0.37mm,\kern0.15mm\{\,\eta :
v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\not=u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\}\kern0.15mm\}
\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ holds. If we
have \mathss38{
v\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.2mm\setminus\kern-0.3mm{{}^{}{\rm dom}\,{}_{{}^{}}} u
\not\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \,we see that
$\loint u\rmdss11\mu < \loint v\rmdss11\mu$ holds, leading to
a contradiction. Similarly we see that \math{\{\,\eta :
v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\not=u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\}
\not\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } is impossible. So we have
\mathss38{v\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.2mm\setminus\kern-0.3mm{{}^{}{\rm dom}\,{}_{{}^{}}} u\cupss31\{\,\eta :
v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\not=u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
The implication \math{(\kern0.15mm*\kern0.15mm)\impss33 0\le\plusint x\rmdss20\mu < \lower1.05mm\hbox{$^+$}\infty }
is straightforward.
\end{proof} | 2,509 | 362,273 | en |
train | 0.47.35 | Assuming that \math{\mu} is a positive measure on \math{{}^{}\Cal Omega} and that \math{
x} is a function \mathss36{{}^{}\Cal Omega\to\mathbb C}, \,from Lemma \ref{Le +int} via
inspection of items (\ref{defi Reint}) and (\ref{defi ps-usual int}) in
Constructions \ref{defi re scal int} above we see that \math{ \smb I =
\int_{\KPp1.1{}^{}\Cal Omega}\kern0.37mm x\rmdss11\mu\not=\hbox{\font\SweD =cmssbx10\SweD U}{} } implies that \math{\smb I
\in\mathbb C} holds together with \math{(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm) \label{int fin a.e. meas} }
being finitely almost \mathss37{\mu}--\,measurable. Thus in the case of an
incomplete probability measure an {\sl integrable function need not be \label{int not meas}
measurable\kern0.15mm} according to our conventions.
\begin{proposition}\label{Pro upint}
Let $\,p\in\rbb R^+$ and
let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega\,$. Also let
$\,w=
\seqss43{u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta + v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta:\eta=\eta}$
where $\,u$ and $\,v$ are any functions with
$\,u\cupss22 v\subseteq{}^{}\Cal Omega\times[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] \KP1 $.
Then $\,\|\,w\,\|\Lnorss33^p_\mu
\le
\sup\KPt8\{\,1\kern0.37mm,\kern0.07mm 2\KP1^{p^{-1}-\kern0.37mm 1\kern0.37mm}\big\}\KP1\big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0$ holds.
\end{proposition}
\begin{proof} Let $\,\roman M\,x=
\kern0.15mm^{}^{}\Cal Omega\KP1[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9]
\capss51\{\,\varphi:\varphi\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.07mm\barscTbb_R\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ \inskipline{.2}{56}
$\text{ and }\kern0.37mm\aall{\eta\,,\kern0.15mm t}\,
(\kern0.37mm\eta\,,\kern0.07mm t\kern0.37mm)\in x\impss33
t\le\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
\KP1\} $ . \inskipline{.4}0
From \cite[pp.\ 49\,--\,50]{Jr} we know that the assertion
holds under the additional restriction that we have
\math{u\in\roman M\,u} and \math{v\in\roman M\,v} with
\math{{}^{}{\rm rng}\,{}_{{}^{}}(\kern0.37mm u\cupss22 v\kern0.37mm)\subseteq\ssbb05 R}, \,noting that
it is trivial if \math{
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu=\lower1.05mm\hbox{$^+$}\infty} holds. From
this one easily extends the result to the case
where the restriction $\lower1.05mm\hbox{$^+$}\infty\not\in{}^{}{\rm rng}\,{}_{{}^{}}(\kern0.37mm u\cupss22 v\kern0.37mm)$ is
removed.
Now putting
\math{\smb A=\sup\KPt8\{\,1\kern0.37mm,\kern0.07mm 2\KP1^{p^{-1}-\kern0.37mm 1\kern0.37mm}\big\} } for
the general case, to proceed indirectly,
suppose that we have
$\smb A\,\big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0
<\|\,w\,\|\Lnorss33^p_\mu$ and take any $\varepsilon\in\rbb R^+$ with
$2\KPt8\smb A\KPt8\varepsilon < \|\,w\,\|\Lnorss33^p_\mu -
\smb A\,\big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0$ .
Then there are $\varphi\in\roman M\,u$ and $\psi\in\roman M\,v$ with
$
\|\,\varphi\,\|\Lnorss33^p_\mu
< \|\,u\,\|\Lnorss33^p_\mu + \varepsilon \,$ and $\,
\|\,\psi\,\|\Lnorss33^p_\mu < \|\,v\,\|\Lnorss33^p_\mu + \varepsilon \,$,
whence with
$\chi=\seqss43{\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta + \psi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta:\eta=\eta}$
we obtain $\chi\in\roman M\,w$ and consequently \inskipline{.7}{8.5}
$\|\,\chi\,\|\Lnorss33^p_\mu\le\smb A\,\big(\kern0.37mm
\|\,\varphi\,\|\Lnorss33^p_\mu + \|\,\psi\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0
$ \inskipline{.4}{21}
${}<\smb A\,\big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0
+2\KPt8\smb A\KPt8\varepsilon
< \|\,w\,\|\Lnorss33^p_\mu \,$, \,a contradiction.
\end{proof} | 1,945 | 362,273 | en |
train | 0.47.36 | \begin{remark}\label{Rem +*}
According to our updated definitional conventions in \cite{Hif} concerning
sums and products of elements in a {\sl pseudo\kern0.37mm-\kern0.15mm usual algebroid\kern0.15mm}, if
in Proposition \ref{Pro upint} for \mathss03{ \scrmt A =
\{\,{{}^{}{\rm dom}\,{}_{{}^{}}} u\kern0.37mm,{{}^{}{\rm dom}\,{}_{{}^{}}} v\,\} } we have \math{\bigcap\,\scrmt A \not= \emptyset }
or \mathss36{\bigcap\,\scrmt A = \emptyset = \bigcup\,\scrmt A }, \,then also \mathss30{w =
u + v } holds. Hence under this additional assumption we could have written
the expression for \math{w} a bit more simply. However, if \math{
\bigcap\,\scrmt A = \emptyset \not= \bigcup\,\scrmt A } holds and we also
have \mathss06{\|\,u\,\|\Lnorss33^p_\mu \not= \lower1.05mm\hbox{$^+$}\infty \not=
\|\,v\,\|\Lnorss33^p_\mu }, \,then \math{u + v = \hbox{\font\SweD =cmssbx10\SweD U}{} } and for this in
place of \math{w} we would \linebreak \vskip-3.3mm\noindentdent get $ \KP{3.3}
\lower1.05mm\hbox{$^+$}\infty = \inf\kern0.37mm\emptyset = \|\KPt8\hbox{\font\SweD =cmssbx10\SweD U}{}\KPt8\|\Lnorss33^p_\mu
= \|\KP1 u + v\KP1\|\Lnorss33^p_\mu = \|\,w\,\|\Lnorss33^p_\mu$ \inskipline{.2}{15}
${} \le \sup\KPt8\{\,1\kern0.37mm,\kern0.07mm 2\KP1^{p^{-1}-\kern0.37mm 1\kern0.37mm}\big\}\KP1\big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0
< \lower1.05mm\hbox{$^+$}\infty \,$, \,a contradiction. \inskipline{.6}0
A similar remark applies to \math{\varphi\,,\kern0.15mm\psi\kern0.37mm,\kern0.15mm\chi} in the proof,
thus having \math{\chi = \varphi + \psi } the function given by \math{{}^{}\Cal Omega
\owns\eta\mapsto\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta + \psi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } since here \math{
{}^{}\Cal Omega={{}^{}{\rm dom}\,{}_{{}^{}}}\varphi={{}^{}{\rm dom}\,{}_{{}^{}}}\psi } holds.
We also suggest the reader to see \cite{A+Mo} for another kind of treatment of
the notational \q{plus\kern0.37mm-\kern0.15mm times} problem referred to above.
\end{remark}
We also extend H\"older's inequality to upper integrals in the next
\begin{proposition}\label{Pro Hˆlder}
Let $\,1\le p < \lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega
\,$. Also let
$\,w=\seqss43{u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\cdot(\kern0.37mm v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm):\eta=\eta}$ where $\,u$
and $\,v$ are any functions with
$\,u\cupss22 v\subseteq{}^{}\Cal Omega\times[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] \KP1 $.
Then $\,\upint w\rmdss11\mu
\le \|\,u\,\|\Lnorss33^p_\mu \KP1 \|\,v\,\|\Lnorss40^{p\sast}_\mu$ holds.
\end{proposition}
\begin{proof} Let \math{\roman M\,x} be as in the proof of Proposition \ref{Pro upint}
above. For the indirect verification, suppose now that we have \mathss37{
\|\,u\,\|\Lnorss33^p_\mu \KP1 \|\,v\,\|\Lnorss40^{p\sast}_\mu <
\upint w\rmdss11\mu}, \,and with \mathss03{ A =
\big\{\,\|\,u\,\|\Lnorss33^p_\mu \,,\kern0.15mm \|\,v\,\|\Lnorss40^{p\sast}_\mu\kern0.37mm\} }
then put \mathss36{\smb M=\sup\,A}. We cannot have \math{0\in A} since by a
simple exercise this would force \mathss36{ \upint w\rmdss11\mu = 0
}, \,contradicting our assumption. It follows that \math{\smb M < \lower1.05mm\hbox{$^+$}\infty}
holds, and we then take any \math{\varepsilon} with \vskip.5mm\centerline{$
0 < \varepsilon < \inf\,\{\,1\kern0.37mm,\kern0.15mm(\kern0.37mm 2\KPt8\smb M + 1\kern0.37mm)\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.37mm
\big(\kern0.15mm\upint w\rmdss11\mu - \|\,u\,\|\Lnorss33^p_\mu \KP1
\|\,v\,\|\Lnorss40^{p\sast}_\mu \kern0.15mm \sbig) 0 \,\big\} \KP1 $.} \inskipline{.5}0
Now there are functions \math{\varphi\in\roman M\,u} and \math{ \psi \in
\roman M\,v} with \math{ \|\,\varphi\,\|\Lnorss33^p_\mu <
\|\,u\,\|\Lnorss33^p_\mu + \varepsilon } and \mathss05{
\|\,\psi\,\|\Lnorss40^{p\sast}_\mu < \|\,v\,\|\Lnorss40^{p\sast}_\mu + \varepsilon
}, \,whence taking \math{ \chi =
\seqss43{\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta \cdot(\kern0.37mm\psi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm):\eta=\eta} } we
then have \math{\chi\in\roman M\,w} and consequently by the usual H\"older's
inequality extended to measurable functions with values in \math{
[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } we obtain \inskipline{.7}{7.78}
$ \upint w\rmdss11\mu \le \int_{\KPp1.1{}^{}\Cal Omega\,}\chi\rmdss11\mu
\le \|\,\varphi\,\|\Lnorss33^p_\mu \KP1 \|\,\psi\,\|\Lnorss40^{p\sast}_\mu $ \inskipline{.4}{20}
${}< \big(\kern0.37mm\|\,u\,\|\Lnorss33^p_\mu + \varepsilon\kern0.37mm)\KP1
\big(\kern0.37mm\|\,v\,\|\Lnorss40^{p\sast}_\mu + \varepsilon\kern0.37mm) $ \inskipline{.4}{20}
${}= \|\,u\,\|\Lnorss33^p_\mu\KP1\|\,v\,\|\Lnorss40^{p\sast}_\mu + \big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss40^{p\sast}_\mu\kern0.37mm)\KP1\varepsilon
+ \varepsilon\KPt8^2 $ \inskipline{.4}{20}
${}\le \|\,u\,\|\Lnorss33^p_\mu\KP1\|\,v\,\|\Lnorss40^{p\sast}_\mu +
(\kern0.37mm 2\KPt8\smb M + 1\kern0.37mm)\KP1\varepsilon
< \upint w\rmdss11\mu \,$, \,a contradiction.
\end{proof} | 2,513 | 362,273 | en |
train | 0.47.37 | \begin{constructions}[standard Lebesgue measures]\label{defi Leb mea} $\null$ \vskip.5mm
\begin{enumerate}\begin{myLeftskip}{-4}{.6}{.4}
\item \ $\upCth\mu = \mu\KP1|\KP1\{\,A:\aall{B\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}\, \label{ctr Carath}
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss21 B\kern0.37mm) +
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm B\kern0.15mm\setminus A\kern0.37mm) \le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm B\KP1\} \KP1 $,
\item \ $\mu\meatimes\nu = \uniqset\mu\ar 1\kern-0.3mm:\aall{\mu\ar 2\kern0.37mm,\kern0.15mm\mu\ar 3}$ \newskline8
$\mu\ar 2 = \{\,(\kern0.07mm A\times B\kern0.37mm,\kern0.15mm s\kern-.2mm\cdot\kern-.2mm t\,):
(\kern0.07mm A\, ,\kern0.07mm s\kern0.37mm)\in\mu \kern0.37mm$ and $\kern0.37mm
(\kern0.07mm B\kern0.37mm,\kern0.07mm t\kern0.37mm)\in\nu\,\}\kern0.37mm$ and \vskip-.3mm
$\nKP{12.7} \mu\ar 3 =
\big\langle\,\inf\kern0.15mm\big\{\,\sum\,(\kern0.37mm\mu\ar 2\circ\ebit B\kern0.37mm)
: \mu\ar 2\kern-0.2mm : \ebit B\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 2)\kern0.37mm$ and \newfline
$ A\subseteq\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\ebit B\KPt8\} : \mu\ar 2\kern-0.2mm :
A\subseteq\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 2\,\big\rangle \impss22
\mu\ar 1 = \upCth\mu\ar 3 \KPt8 $, \KP{8.5}
\item \ $\Lebmef^{\kern0.37mm\ssmb N} = \uniqset\mu\ar 1\kern-0.3mm:\smb N\in\mathbb No\kern0.37mm$ and $\, \label{ctr Lebmea}
[\ [\KP{1.3}\smb N = \emptyset\kern0.37mm$ and $\kern0.37mm
\mu\ar 1 = \seq{\,0\,,1\,} \KP{1.3}]\kern0.37mm$ or \newskline{15}
$[\KP{1.3} \smb N \not= \emptyset\kern0.37mm$ and $\,\aall{
\Cal B\kern0.37mm,\kern0.15mm\Cal J\kern0.07mm,\kern0.07mm\mu\,,\kern0.07mm\nu\kern0.37mm,\kern0.07mm\nu\ar 1}\,
\Cal J = \{\,\openIval{\smb A\kern0.37mm,\smb B} :
\smb A\kern0.37mm,\kern0.07mm\smb B\in\ssbb09 R\}\kern0.37mm$ and \newskline{6.5}
$\nu\ar 1 = \seq{\,\smb B-\smb A : J = \openIval{\smb A\kern0.37mm,\smb B}
\in\Cal J\kern0.37mm$ and $\kern0.37mm\smb A\le\smb B\,}\kern0.37mm$ and \newskline{8}
$\nu = \Seq{\,\prod\,(\kern0.37mm\nu\ar 1\kern-0.2mm\circ\kern0.07mm\ebit I\kern0.37mm) : B =
\prod{_{_{\kern-.3mm\bold c\kern.15mm}}}\ebit I\kern0.37mm$ and $\kern0.37mm\ebit I\in\kern0.15mm\yi N\kern0.15mm\Cal J\KP1}\kern0.37mm$ and $\kern0.37mm
\Cal B={{}^{}{\rm dom}\,{}_{{}^{}}}\nu$
$\nKP{8} \mu = \Seq{\,\inf\kern0.15mm\big\{\,\sum\,(\kern0.37mm\nu\circ\ebit B\kern0.37mm) :
\ebit B\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\Cal B\kern0.37mm$ and $\kern0.37mm A\subseteq\bigcup\kern0.15mm{}^{}{\rm rng}\,{}_{{}^{}}\ebit B\,\} :
A\subseteq\kern0.15mm\yi N\ssbb69 R}$ \newfline
$\impss02 \mu\ar 1 = \upCth\mu \KP{1.3}]\ ]\ $, \KP{8.5}
\item \ $\Lebmef^{} = \Lebmef^{\kern0.15mm 1.} \circ\kern0.15mm
\big\langle\KP1^{1.}A:A\subseteq\ssbb09 R\kern0.15mm\rangle \KP1 $, \label{def Lebm on R} \vskip.3mm
\item \ $\int_{\,\ssmb A}^{\,\ssmb B\,}x = \uniqset\smb I : \label{df Leb int}
\eexi{A\,,\kern0.07mm\sigma}\,\big[\ \big[\KP{1.5} \sigma = 1\kern0.37mm$ and $\kern0.37mm
\smb A < \smb B\kern0.37mm$ and $\kern0.37mm
A = \openIval{\smb A\kern0.37mm,\kern0.07mm\smb B} \KP{1.5} \big]$ \inskipline{-.1}{43.5}
or $\kern0.37mm\big[\KP{1.5} \sigma = \kern.2mm\lower1.05mm\hbox{$^-$} 1\kern0.37mm$ and $\kern0.37mm \smb B \le \smb A\kern0.37mm$
and $\kern0.37mm A = \openIval{\smb B\kern0.37mm,\kern0.07mm\smb A} \KP{1.5} \big]\ \big]$ \inskipline{-.1}{44.33}
and $\, \smb I = \sigma\kern0.07mm\int_{\,A}\kern0.37mm x\rmdss10\Lebmef^{} \KP1 $.
\end{myLeftskip}\end{enumerate}
\end{constructions}
\newcommand\sPows{{\lower.22mm\hbox{\font\≈=eusm8\≈P}\kern-.3mm\lower.7mm\hbox{\font\≈=cmss5\≈s}\kern.65mm}}
Saying that \math{\mu} is an {\it outer measure\kern0.37mm} on \math{{}^{}\Cal Omega} if{}f \math{
\mu\in\kern0.15mm^{\sPows{}^{}\Cal Omega}\KP1[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } and \mathss30{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\le\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm B } and \math{
\mu\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\bigcup\,\scrmt A\le\sum\KP1(\kern0.37mm\mu\KP1|\KP1\scrmt A\kern0.37mm) }
hold whenever we have \math{A\subseteq B\subseteq{}^{}\Cal Omega } and \math{ \scrmt A \subseteq
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm{}^{}\Cal Omega } with \linebreak
$\scrmt A\kern0.37mm$ countable, essentially from
\cite[Lemma 3.1.8\kern0.37mm, Proposition 3.1.9\kern0.37mm, pp.\ 67\,--\,68]{Du} we get the
proof of the following
\begin{proposition}\label{Pro Cth}
If $\,\mu$ is an outer measure on $\,{}^{}\Cal Omega${\,\rm, }then $\,\upCth\mu$ is a
complete positive measure on $\,{}^{}\Cal Omega\,$.
\end{proposition}
Thus by Proposition \ref{Pro Cth} in \ref{defi Leb mea}\,(\ref{ctr Carath}) we
have the standard {\sl Carath\'eodory construc- tion\kern0.15mm} associating a
complete positive measure with any outer measure. For \mathss36{\smb N\in\mathbb N
}, \,the function \math{\Lebmef^{\kern0.37mm\ssmb N}} is the standard complete
Lebesgue measure on \math{\yi N\ssbb60 R} defined on the class of Lebesgue
measurable subsets. The corresponding measure on \math{\mathbb R} is \math{
\Lebmef^{}\kern0.15mm}. Note that if we had not separately defined \math{
\Lebmef^{\kern0.15mm 0.} = \seqss20{0\,,\kern0.07mm 1} = \{\,(\kern0.37mm\emptyset\,,\kern0.07mm 0\kern0.37mm)\,,\kern0.15mm
(\kern0.37mm 1\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.07mm 1\kern0.37mm)\,\} } by inserting \q{\mathss03{ \smb N =
\emptyset\kern0.37mm\text{ and }\kern0.37mm\mu\ar 1=\seqss20{0\,,\kern0.07mm 1} }} in
\ref{defi Leb mea}\,(\ref{ctr Lebmea})\,, then it would have given \mathss30{
\Lebmef^{\kern0.15mm 0.}=\kern-0.3mm} \mathss08{2\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-.2mm\times\kern-.2mm\{\kern0.15mm\lower1.05mm\hbox{$^+$}\infty\,\} }. We also
put \math{ \LeBmef^{\kern0.37mm\ssmb N} =
\Lebmef^{\kern0.37mm\ssmb N}\,|\KP1\sigmAlg3\nsTbb_R\kern-0.63mm\yi N } and \math{\LeBmef^{}
= \Lebmef^{}\,|\KP1\sigmAlg3\nsTbb_R } get- ting the restrictions of the
Lebesgue measures to the standard Borel \rsigma3algebras.
\insubsubhead Pettis integration of vector functions \label{Ss Pettis}
In some places of the proof of Theorem \nfss A\,\ref{main Th} we refer to
something being {\sl Pettis\kern0.15mm}. In order to make the meaning of this
explicit, we give the following
\begin{definitions}[for Pettis integration]\label{df Petti}
(1) \ Say that \math{\tilde c} is {\it scalar integrable\kern0.37mm} to \math{x}
if{}f \math{\tilde c} is an mv\kern0.37mm-\kern0.15mm map and for all \math{\bosy K\kern0.37mm,\kern0.15mm
E\kern0.37mm,\kern0.15mm\mu\,,\kern0.15mm{}^{}\Cal Omega\kern0.37mm,\kern0.15mm x\kern0.37mm,\kern0.15mm u} from \math{ \tilde c =
(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm E\kern0.37mm) } and \math{{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{{{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K } and \math{ u \in
\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } it follows that \newline
$u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x=\int_{\KPp1.1{}^{}\Cal Omega}\,u\circss01 c\rmdss21\mu\not=\hbox{\font\SweD =cmssbx10\SweD U}{}$ holds, \inskipline{.5}2 | 4,008 | 362,273 | en |
train | 0.47.38 | \begin{proposition}\label{Pro Cth}
If $\,\mu$ is an outer measure on $\,{}^{}\Cal Omega${\,\rm, }then $\,\upCth\mu$ is a
complete positive measure on $\,{}^{}\Cal Omega\,$.
\end{proposition}
Thus by Proposition \ref{Pro Cth} in \ref{defi Leb mea}\,(\ref{ctr Carath}) we
have the standard {\sl Carath\'eodory construc- tion\kern0.15mm} associating a
complete positive measure with any outer measure. For \mathss36{\smb N\in\mathbb N
}, \,the function \math{\Lebmef^{\kern0.37mm\ssmb N}} is the standard complete
Lebesgue measure on \math{\yi N\ssbb60 R} defined on the class of Lebesgue
measurable subsets. The corresponding measure on \math{\mathbb R} is \math{
\Lebmef^{}\kern0.15mm}. Note that if we had not separately defined \math{
\Lebmef^{\kern0.15mm 0.} = \seqss20{0\,,\kern0.07mm 1} = \{\,(\kern0.37mm\emptyset\,,\kern0.07mm 0\kern0.37mm)\,,\kern0.15mm
(\kern0.37mm 1\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.07mm 1\kern0.37mm)\,\} } by inserting \q{\mathss03{ \smb N =
\emptyset\kern0.37mm\text{ and }\kern0.37mm\mu\ar 1=\seqss20{0\,,\kern0.07mm 1} }} in
\ref{defi Leb mea}\,(\ref{ctr Lebmea})\,, then it would have given \mathss30{
\Lebmef^{\kern0.15mm 0.}=\kern-0.3mm} \mathss08{2\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-.2mm\times\kern-.2mm\{\kern0.15mm\lower1.05mm\hbox{$^+$}\infty\,\} }. We also
put \math{ \LeBmef^{\kern0.37mm\ssmb N} =
\Lebmef^{\kern0.37mm\ssmb N}\,|\KP1\sigmAlg3\nsTbb_R\kern-0.63mm\yi N } and \math{\LeBmef^{}
= \Lebmef^{}\,|\KP1\sigmAlg3\nsTbb_R } get- ting the restrictions of the
Lebesgue measures to the standard Borel \rsigma3algebras.
\insubsubhead Pettis integration of vector functions \label{Ss Pettis}
In some places of the proof of Theorem \nfss A\,\ref{main Th} we refer to
something being {\sl Pettis\kern0.15mm}. In order to make the meaning of this
explicit, we give the following
\begin{definitions}[for Pettis integration]\label{df Petti}
(1) \ Say that \math{\tilde c} is {\it scalar integrable\kern0.37mm} to \math{x}
if{}f \math{\tilde c} is an mv\kern0.37mm-\kern0.15mm map and for all \math{\bosy K\kern0.37mm,\kern0.15mm
E\kern0.37mm,\kern0.15mm\mu\,,\kern0.15mm{}^{}\Cal Omega\kern0.37mm,\kern0.15mm x\kern0.37mm,\kern0.15mm u} from \math{ \tilde c =
(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm E\kern0.37mm) } and \math{{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{{{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K } and \math{ u \in
\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } it follows that \newline
$u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x=\int_{\KPp1.1{}^{}\Cal Omega}\,u\circss01 c\rmdss21\mu\not=\hbox{\font\SweD =cmssbx10\SweD U}{}$ holds, \inskipline{.5}2
(2) \ $E\vPettis3int_A\,c\rmdss21\mu=\uniqset x:c\kern0.37mm$ a function and \math{
\mu} is a positive measure and \inskipline{.2}{8.5}
$A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} c\capss43\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ and \math{
(\,c\KP{1.2}|\KP1 A\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm E\kern0.37mm)} is scalar
integrable to \mathss34{x}, \inskipline{.5}2
(3) \ Say that \math{\tilde c} is {\it Pettis\kern0.37mm} if{}f
\math{\tilde c} is an mv\kern0.37mm-\kern0.15mm map and for all
\math{A\,,\kern0.15mm E\kern0.37mm,\kern0.15mm\mu\,,\kern0.15mm c} from
$\kern0.37mm\tilde c={}$ \inskipline{.2}{8.5}
$(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm E\kern0.37mm) \kern0.37mm$ and
\math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} it follows that \math{
E\vPettis3int_A\,c\rmdss21\mu\not=\hbox{\font\SweD =cmssbx10\SweD U}{} } holds.
\end{definitions}
Because of the manner we have put the definitions, from the discussion after
the proof of Lemma \ref{Le +int} on page \pageref{int not meas} above, it
follows that \math{\tilde c} being Pettis implies it being {\sl finitely
almost scalarly measurable\kern0.15mm}. Also from \math{(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm E\kern0.37mm) }
being Pettis with \mathss03{{}^{}{\rm rng}\,{}_{{}^{}}\mu\capss34\rbb R^+\not=\emptyset} it follows
that \math{{}^{}{\rm Card}{}^{}\,\kern0.15mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\not=1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}={}^{}{\rm Card}{}^{}\,\kern0.15mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm(\kern0.15mm E\dlsigss11\kern0.15mm) }
cannot hold. That is, if \math{\mu} and \math{E} are nontrivial, then also the
dual of \math{E} must be such. For example \mathss03{
(\,c\KPt8;\Lebmef^{}\kern0.37mm,\kern0.07mm\LLrs42^{\frac 12}(\ssbb44 I)) } cannot be
Pettis, whereas \math{
(\,c\KPt8;\Lebmef^{}\kern0.37mm,\kern0.15mm\ell\KPt8^{\frac 12\kern0.37mm}(\kern0.37mm\mathbb No\kern0.07mm)) } can. \vskip.3mm
To have at our disposal also some partially weaker and more general notions of
integrability of mv\kern0.37mm-\kern0.15mm maps, we put the following
\begin{definitions}\label{df sc+Gel-int}
(1) \ Say that \math{\tilde c} is {\it scalarly integrable\kern0.37mm} if{}f \math{
\tilde c} is an mv\kern0.37mm-\kern0.15mm map and for all \mathss03{c\,,\kern0.15mm\mu\,,\kern0.15mm
\bosy K\kern0.15mm,\kern0.15mm\vPi} from \math{\tilde c=(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K={{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi } it follows that
for all \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus\{\,\Bnull_\vPi\} } there is \math{
u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } with \math{u\fvalss01\xi\not=0 }
and for all \math{u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } and \math{ A \in
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} it holds that \mathss36{
\int_{\,A\,}u\circss01 c\rmdss21\mu \not= \hbox{\font\SweD =cmssbx10\SweD U}{} }, \inskipline{.5}2
(2) \ Say that \math{\tilde c} is {\it finitely scalarly integrable\kern0.37mm} if{}f \math{
\tilde c} is an mv\kern0.37mm-\kern0.15mm map and for all \mathss30{c\,,\kern0.15mm\mu\,,} $
\vPi\kern0.37mm$ from \math{\tilde c=(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+} it follows that \math{
(\,c\KPp1.2|\KP1 A\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\kern0.37mm) } is scalarly
integrable, \inskipline{.5}2
(3) \ Say that \math{\tilde c} is {\it Gelfand\,} if{}f \math{\tilde c} is
scalarly integrable and for all \mathss30{c\,,\kern0.15mm\mu\,,\kern0.15mm A\,,\kern0.15mm\bosy K\kern0.15mm
,\kern0.15mm\vPi} from \math{\tilde c=(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K={{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi } and \mathss30{A\in
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} it follows that \math{
\big\langle\kern0.15mm\int_{\,A\,}u\circss01 c\rmdss21\mu : u\in
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)\KP1\rangle } is continuous \mathss32{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm)\to\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}.
\end{definitions}
A simple example of a Banach space valued mv\kern0.37mm-\kern0.15mm map that is Gelfand but
not Pettis is given in the following
\begin{example}\label{Exa Gel /= Pet} | 4,052 | 362,273 | en |
train | 0.47.39 | (2) \ Say that \math{\tilde c} is {\it finitely scalarly integrable\kern0.37mm} if{}f \math{
\tilde c} is an mv\kern0.37mm-\kern0.15mm map and for all \mathss30{c\,,\kern0.15mm\mu\,,} $
\vPi\kern0.37mm$ from \math{\tilde c=(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+} it follows that \math{
(\,c\KPp1.2|\KP1 A\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\kern0.37mm) } is scalarly
integrable, \inskipline{.5}2
(3) \ Say that \math{\tilde c} is {\it Gelfand\,} if{}f \math{\tilde c} is
scalarly integrable and for all \mathss30{c\,,\kern0.15mm\mu\,,\kern0.15mm A\,,\kern0.15mm\bosy K\kern0.15mm
,\kern0.15mm\vPi} from \math{\tilde c=(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K={{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi } and \mathss30{A\in
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} it follows that \math{
\big\langle\kern0.15mm\int_{\,A\,}u\circss01 c\rmdss21\mu : u\in
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)\KP1\rangle } is continuous \mathss32{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm)\to\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}.
\end{definitions}
A simple example of a Banach space valued mv\kern0.37mm-\kern0.15mm map that is Gelfand but
not Pettis is given in the following
\begin{example}\label{Exa Gel /= Pet}
Let \math{\tilde x=(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } where \math{\mu=
\seqss30{\card3 A:A\subseteq\mathbb No\kern-0.2mm} } and \math{\vPi=\hbox{\font\≈=cmmi12\≈c}\lower.8mm\hbox{\font\≈=cmr6\≈o}\kern.4mm(\kern0.37mm\mathbb No\kern0.07mm) }
and \mathss08{x=\seqss30{(\kern0.37mm\mathbb No\kern-0.3mm\setminus\{\kern0.37mm i\kern0.37mm\}\kern0.15mm\sbig)0\times\kern-0.2mm
\{\kern0.37mm 0\kern0.37mm\}\cupss22\{\,(\kern0.37mm i\kern0.37mm,\kern0.07mm 1\kern0.37mm)\,\}:i\in\mathbb No\kern-0.2mm} }. For
every \math{A\subseteq\mathbb No } and \mathss30{ \zeta \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\ell\KPt8^1\kern0.07mm(\kern0.37mm\mathbb No\kern0.07mm) } then \mathss38{
\int_{\,A\,}x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\cdot\zeta\rmdss21\mu\,(\kern0.15mm\eta\kern0.15mm) =
\sum\KP1(\kern0.37mm\zeta\KP1|\KP1 A\kern0.37mm) }, \,and hence \math{\tilde x} is scalarly
integrable. It is also Gelfand since for \math{ \lambda =
\mathbb No\kern-.2mm\times\kern-.2mm\{\kern0.37mm 1\kern0.37mm\} } we have \math{ \lambda \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.07mm(\kern0.37mm\mathbb No\kern0.07mm) } with \mathss30{
\sum\KP1(\kern0.37mm\lambda\cdot\zeta\KP1|\KP1 A\kern0.37mm) = \kern-0.3mm} \mathss03{
\sum\KP1(\kern0.37mm\zeta\KP1|\KP1 A\kern0.37mm) =
\int_{\,A\,}x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\cdot\zeta\rmdss21\mu\,(\kern0.15mm\eta\kern0.15mm) } for all \math{
A\subseteq\mathbb No} and \mathss38{\zeta\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\ell\KPt8^1\kern0.07mm(\kern0.37mm\mathbb No\kern0.07mm) }. Since
we here have \mathss31{\lambda\not\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi}, \,we see that \math{\tilde x}
is not Pettis.
\end{example}
\Ssubhead B Generalized Bochner spaces \label{Sec B}
In this section, we first give the formal construction of the generalized
Lebesgue\,--\,Bochner spaces spaces \math{ F =
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{ F\aar 1 =
\mvsLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } of equivalence classes of order \math{p}
integrable functions \math{x:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} when \math{ 0 \le p \le
\lower1.05mm\hbox{$^+$}\infty} and \math{\mu} is a positive measure on some set \math{{}^{}\Cal Omega}
and \math{\vPi} is a real or complex topological vector space. Then we
establish the basic relevant properties of these spaces under the additional
assumption that the space \math{\vPi} is suitable.
For the construction of the space $\kern0.37mm F$, the functions \math{x} are
required to be such that \math{\tilde x=(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm)} is
{\sl finitely almost simply measurable\kern0.15mm} in the sense of
Definitional schemata \ref{df meas} on page \pageref{df meas} above. For \math{
F\aar 1} we instead require \math{\tilde x} to be only {\sl finitely almost
scalarly measurable\kern0.15mm}.
The integrability condition is formulated so that in the case \math{p\not=0}
for any bounded quasi\kern0.37mm-\kern0.37mm seminorm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} in \math{\vPi} we should
have \math{\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu<\lower1.05mm\hbox{$^+$}\infty } which in the
case \math{p\in\rbb R^+} is equivalent to the function \math{
\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 x : {}^{}\Cal Omega \owns \eta \mapsto
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 x\fvalss11\eta\kern0.37mm)\,^p } \nolinebreak pos- sessing a
dominating \mathss37{
\mu}--\,integrable function \mathss38{ \varphi : {}^{}\Cal Omega \to
[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] }. Then
$\null$
\math{
\upint\kern0.37mm\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 x\rmdss11\mu<\lower1.05mm\hbox{$^+$}\infty} holds, and
this determines
by $x\mapsto\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu$ a corresponding
quasi\kern0.37mm-\kern0.37mm seminorm
.
For \math{p=0} no integrability is required, and in this case the topology is
determined by the quasi\kern0.37mm-\kern0.37mm semimetrics \math{ \roman d\,A\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm :
(\kern0.15mm x\kern0.37mm,\kern0.15mm y\kern0.37mm)\mapsto\upint\kern0.37mm\rmmd\circ\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 z\rmdss11\mu}
where with given \math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+} we have \mathss38{
z = (\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\,\Bnull_\vPi\}\cupss22
\seqss33{(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta-y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\svs\vPi\kern-0.2mm:\eta\in A} }, \,and \math{
\rmmd = \seqss30{ (\kern0.37mm 1 + t\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,t : t \in \lbb R_+ \kern-0.3mm } }
hence \math{\lbb R_+\to[\KPp1.1 0\,,\kern0.07mm 1\KPt9{[\,} } given by \mathss34{
t \mapsto (\kern0.37mm 1 + t\kern0.37mm)^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,t }.
To get Hausdorff topologies for the spaces \math{F} and \mathss32{F\aar 1
}, \,we finally take the quotient space by the vector subspace \math{N\aar 0}
of functions \math{x} with \math{\int_{\,A}\kern0.15mm u\circss01 x\rmdss11\mu=0} for
all \linebreak
$A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.15mm$ and for all \mathss38{
u\in\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) }. | 3,952 | 362,273 | en |
train | 0.47.40 | The integrability condition is formulated so that in the case \math{p\not=0}
for any bounded quasi\kern0.37mm-\kern0.37mm seminorm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} in \math{\vPi} we should
have \math{\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu<\lower1.05mm\hbox{$^+$}\infty } which in the
case \math{p\in\rbb R^+} is equivalent to the function \math{
\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 x : {}^{}\Cal Omega \owns \eta \mapsto
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 x\fvalss11\eta\kern0.37mm)\,^p } \nolinebreak pos- sessing a
dominating \mathss37{
\mu}--\,integrable function \mathss38{ \varphi : {}^{}\Cal Omega \to
[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] }. Then
$\null$
\math{
\upint\kern0.37mm\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 x\rmdss11\mu<\lower1.05mm\hbox{$^+$}\infty} holds, and
this determines
by $x\mapsto\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu$ a corresponding
quasi\kern0.37mm-\kern0.37mm seminorm
.
For \math{p=0} no integrability is required, and in this case the topology is
determined by the quasi\kern0.37mm-\kern0.37mm semimetrics \math{ \roman d\,A\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm :
(\kern0.15mm x\kern0.37mm,\kern0.15mm y\kern0.37mm)\mapsto\upint\kern0.37mm\rmmd\circ\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 z\rmdss11\mu}
where with given \math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+} we have \mathss38{
z = (\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\,\Bnull_\vPi\}\cupss22
\seqss33{(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta-y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\svs\vPi\kern-0.2mm:\eta\in A} }, \,and \math{
\rmmd = \seqss30{ (\kern0.37mm 1 + t\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,t : t \in \lbb R_+ \kern-0.3mm } }
hence \math{\lbb R_+\to[\KPp1.1 0\,,\kern0.07mm 1\KPt9{[\,} } given by \mathss34{
t \mapsto (\kern0.37mm 1 + t\kern0.37mm)^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,t }.
To get Hausdorff topologies for the spaces \math{F} and \mathss32{F\aar 1
}, \,we finally take the quotient space by the vector subspace \math{N\aar 0}
of functions \math{x} with \math{\int_{\,A}\kern0.15mm u\circss01 x\rmdss11\mu=0} for
all \linebreak
$A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.15mm$ and for all \mathss38{
u\in\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) }.
Observe that if with \math{\bosy K=\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R} we have for example \mathss38{\vPi=
\LLrs42^{\frac 12}(\ssbb44 I) }, \,then the dual set \math{
\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) =
\{\KPt8\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern-.2mm\times\kern-.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} } and hence the spaces \math{F}
and \math{F\aar 1 \label{triv L^p exa}} become trivial. For $ \vPi =
\ell\KPt8^{\frac 12}\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm) \kern0.37mm $ the situation is different
since then \math{\vPi} has nontrivial dual.
\begin{constructions}[of generalized Lebesgue\,--\,Bochner spaces]\label{defi $L^p$} $\null$ \vskip.5mm
\begin{enumerate}\begin{myLeftskip}{-4}{.6}{.6}
\item \ $\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
\varXi\sbi M=\uniqset\Cal V:0\le p\le\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and \math{ \label{L^p nbhs}
\eexi{\bosy K\kern0.37mm,\kern0.15mm\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm\vPi} } \newskline{20}
$\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\} \kern0.37mm$ and $\kern0.37mm
\mu\kern0.37mm$ is a positive measure on $\kern0.37mm{}^{}\Cal Omega \kern0.37mm$ and \newfline
$ \vPi\in\tvsps0(K)\kern0.37mm$ and $\kern0.37mm \varXi=(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm$
and $ [ \KP1 [ \KP{1.4} p = 0 \kern0.37mm$ and \KP{11.1} \newskline6
$\Cal V=\{\KPt8 V\kern-0.63mm:\eexi{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm,\kern0.07mm A\,,\kern0.07mm\varepsilon}\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi \kern0.37mm$
and $\kern0.37mm A\in\mu\invss23\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+\kern0.37mm$
and $\kern0.37mm \varepsilon\in\rbb R^+ \kern0.37mm$ and \newfline
$V=M\capss21\{\,x:
\upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1|\KP1 A\rmdss41\mu<\varepsilon\,\}
\kern0.15mm\} \KP{1.4} ] \kern0.37mm$ or $\kern0.37mm [ \KP{1.4} p\in\rbb R^+\kern0.37mm$ and \KP{8.3} \newskline6
$\Cal V=\{\KPt8 V\kern-0.63mm:\eexi{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi}\,V = M\capss21\{\, x :
\upint\kern0.37mm \Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\rmdss41\mu<1\,\}\kern0.15mm\} \KP{1.4} ] $ \newfline
or $\kern0.37mm [ \KP{1.4} p = \lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and \KP7 \newskline6
$\Cal V=\{\KPt8 V\kern-0.63mm:\eexi{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi}\, V = M\capss21\{\,x:
\aall{A\in\mu\invss23\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+}$ \newskline{16}
$\eexi{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }\,
\sup\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1 [\KP1 A\setminus N\KP1]\kern0.37mm) < 1 \KPt8\}\,\}
\KP{1.4} ] \KP{1.2} ] \KP{1.4} $,
\item \ $\raise1.7mm\hbox{\font\≈=cmssi5\≈pr}\kern-.3mm
\LLrs02^p\varXi\kern0.15mm\sbi{M\kern0.37mm\aars N_0}=\uniqset F: \label{preL^p_{MN_0}}
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
\varXi\sbi M\not=\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm$ and \newskline6
$
\aall{\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm\vPi\kern0.15mm,\kern0.15mm S\kern0.37mm,\kern0.15mm\scrmt T\kern0.15mm,\kern0.15mm\Cal V\kern0.15mm,
\kern0.15mm\scrmt V\aar 0\,,\kern0.15mm X\kern0.15mm,\kern0.15mm Y}\,
\varXi=(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm$ and $\kern0.37mm{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$
and \newskline6
$X=\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs}\kern0.15mm$ and $\kern0.37mm\Cal V=
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
\varXi\sbi M\kern0.15mm$ and \newskline6
$S = \kern0.37mm
\bigcap\kern0.37mm\big\{\KP{1.1}[\KP{1.2}\mathbb Z\KP{1.1}V\KP{1.1}]\vvs X\kern-0.3mm:
V\in\Cal V\KP1\} \kern0.37mm$ and $\kern0.37mm Y=
X_{\kern0.37mm|\,S}\,/\vsquotient N\aar 0\kern0.37mm$ and \newskline6 | 4,018 | 362,273 | en |
train | 0.47.41 | $V=M\capss21\{\,x:
\upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1|\KP1 A\rmdss41\mu<\varepsilon\,\}
\kern0.15mm\} \KP{1.4} ] \kern0.37mm$ or $\kern0.37mm [ \KP{1.4} p\in\rbb R^+\kern0.37mm$ and \KP{8.3} \newskline6
$\Cal V=\{\KPt8 V\kern-0.63mm:\eexi{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi}\,V = M\capss21\{\, x :
\upint\kern0.37mm \Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\rmdss41\mu<1\,\}\kern0.15mm\} \KP{1.4} ] $ \newfline
or $\kern0.37mm [ \KP{1.4} p = \lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and \KP7 \newskline6
$\Cal V=\{\KPt8 V\kern-0.63mm:\eexi{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi}\, V = M\capss21\{\,x:
\aall{A\in\mu\invss23\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+}$ \newskline{16}
$\eexi{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }\,
\sup\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1 [\KP1 A\setminus N\KP1]\kern0.37mm) < 1 \KPt8\}\,\}
\KP{1.4} ] \KP{1.2} ] \KP{1.4} $,
\item \ $\raise1.7mm\hbox{\font\≈=cmssi5\≈pr}\kern-.3mm
\LLrs02^p\varXi\kern0.15mm\sbi{M\kern0.37mm\aars N_0}=\uniqset F: \label{preL^p_{MN_0}}
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
\varXi\sbi M\not=\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm$ and \newskline6
$
\aall{\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm\vPi\kern0.15mm,\kern0.15mm S\kern0.37mm,\kern0.15mm\scrmt T\kern0.15mm,\kern0.15mm\Cal V\kern0.15mm,
\kern0.15mm\scrmt V\aar 0\,,\kern0.15mm X\kern0.15mm,\kern0.15mm Y}\,
\varXi=(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm$ and $\kern0.37mm{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$
and \newskline6
$X=\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs}\kern0.15mm$ and $\kern0.37mm\Cal V=
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
\varXi\sbi M\kern0.15mm$ and \newskline6
$S = \kern0.37mm
\bigcap\kern0.37mm\big\{\KP{1.1}[\KP{1.2}\mathbb Z\KP{1.1}V\KP{1.1}]\vvs X\kern-0.3mm:
V\in\Cal V\KP1\} \kern0.37mm$ and $\kern0.37mm Y=
X_{\kern0.37mm|\,S}\,/\vsquotient N\aar 0\kern0.37mm$ and \newskline6
$\scrmt V\aar 0 = \{\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mms Y\capss01\{\,\smb X : \smb X\capss02 V\aar 1
\not= \emptyset \KP1 \} : V\aar 1\in\Cal V\KP1\} \kern0.37mm $ and \newskline6
$\scrmt T=
\{\KPt8 U\kern-0.3mm:\aall{\smb X\in U}\,\eexi{V\in\scrmt V\aar 0}\,
[\KP{1.1}\{\kern0.37mm\smb X\kern0.37mm\} + V\KP{1.2}]\vvs Y\subseteq U\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm Y\KP1\}$ \newskline6
$
\impss33 M\kern0.37mm$ is a vector subspace in $\kern0.37mm X\kern0.37mm$ and \newskline6
$N\aar 0\kern0.37mm$ is a vector subspace in $\kern0.37mm X_{\kern0.37mm|\,S}\kern0.37mm$ and
$\kern0.37mm F=
(\kern0.37mm Y\kern0.07mmp,\kern0.07mm\scrmt T\,) \KPt9 $,
\item \ $\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)=\uniqset F: \eexi{\bosy K}\, \label{simpL^p}
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}\kern0.37mm$ and $\kern0.37mm\vPi\in\tvsps0(K)\kern0.37mm$ and \newskline{11}
$\aall{M\kern0.15mm,\kern0.15mm N\aar 0\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm S\kern0.37mm,\kern0.15mm
\Cal V\kern0.15mm,\kern0.15mm X}\,{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ and $\kern0.37mm
X = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs}\kern0.15mm$ and \newskline{14}
$ M = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm X\capss01\{\,x : (\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm $ is
finitely \newskline{50.5}
almost simply measurable $ \} \kern0.37mm $ and \newskline6
$\Cal V=
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi M\kern0.15mm$ and $\kern0.37mm S = \kern0.37mm
\bigcap\kern0.37mm\big\{\KP{1.1}[\KP{1.2}\mathbb Z\KP{1.1}V\KP{1.1}]\vvs X\kern-0.3mm:
V\in\Cal V\KP1\}\kern0.37mm$ and \newskline6
$N\aar 0=S\capss21\{\,x:\aall{A\,,\kern0.15mm u}\,\eexi N\,A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.15mm$ and $\kern0.37mm u\in
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ \newskline{30}
$\impss03
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.37mm$ and $\kern0.37mm
u\circ x\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.15mm A\setminus N\kern0.37mm)\subseteq\{\kern0.37mm 0\kern0.37mm\}\,\} $ \newfline
$\impss03 \Cal V\not=\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm$ and $\kern0.37mm F=
\raise1.8mm\hbox{\font\≈=cmssi5\≈pr}\kern-.3mm
\LLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi{M\kern0.37mm\aars N_0} \,$, \KP9
\item \ $\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)=\uniqset F: \eexi{\bosy K}\, \label{ctr mvL_s^p}
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}\kern0.37mm$ and $\kern0.37mm\vPi\in\tvsps0(K)\kern0.37mm$ and \newskline{11}
$\aall{M\kern0.15mm,\kern0.15mm N\aar 0\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm S\kern0.37mm,\kern0.15mm
\Cal V\kern0.15mm,\kern0.15mm X}\,{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ and $\kern0.37mm
X = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs}\kern0.15mm$ and \newskline{14}
$ M = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm X\capss01\{\,x : (\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm $ is
finitely \newskline{49}
almost scalarly measurable $ \} \kern0.37mm $ and \newskline6
$\Cal V=
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi M\kern0.15mm$ and $\kern0.37mm S = \kern0.37mm
\bigcap\kern0.37mm\big\{\KP{1.1}[\KP{1.2}\mathbb Z\KP{1.1}V\KP{1.1}]\vvs X\kern-0.3mm:
V\in\Cal V\KP1\}\kern0.37mm$ and \newskline6
$N\aar 0=S\capss21\{\,x:\aall{A\,,\kern0.15mm u}\,\eexi N\,A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.15mm$ and $\kern0.37mm u\in
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ \newskline{30} | 4,002 | 362,273 | en |
train | 0.47.42 | $N\aar 0=S\capss21\{\,x:\aall{A\,,\kern0.15mm u}\,\eexi N\,A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.15mm$ and $\kern0.37mm u\in
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ \newskline{30}
$\impss03
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.37mm$ and $\kern0.37mm
u\circ x\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.15mm A\setminus N\kern0.37mm)\subseteq\{\kern0.37mm 0\kern0.37mm\}\,\} $ \newfline
$\impss03 \Cal V\not=\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm$ and $\kern0.37mm F=
\raise1.8mm\hbox{\font\≈=cmssi5\≈pr}\kern-.3mm
\LLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi{M\kern0.37mm\aars N_0} \,$, \KP9
\item \ $\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)=\uniqset F: \eexi{\bosy K}\, \label{ctr mvL_s^p}
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}\kern0.37mm$ and $\kern0.37mm\vPi\in\tvsps0(K)\kern0.37mm$ and \newskline{11}
$\aall{M\kern0.15mm,\kern0.15mm N\aar 0\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm S\kern0.37mm,\kern0.15mm
\Cal V\kern0.15mm,\kern0.15mm X}\,{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ and $\kern0.37mm
X = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs}\kern0.15mm$ and \newskline{14}
$ M = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm X\capss01\{\,x : (\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm $ is
finitely \newskline{49}
almost scalarly measurable $ \} \kern0.37mm $ and \newskline6
$\Cal V=
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi M\kern0.15mm$ and $\kern0.37mm S = \kern0.37mm
\bigcap\kern0.37mm\big\{\KP{1.1}[\KP{1.2}\mathbb Z\KP{1.1}V\KP{1.1}]\vvs X\kern-0.3mm:
V\in\Cal V\KP1\}\kern0.37mm$ and \newskline6
$N\aar 0=S\capss21\{\,x:\aall{A\,,\kern0.15mm u}\,\eexi N\,A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.15mm$ and $\kern0.37mm u\in
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ \newskline{30}
$\impss03
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.37mm$ and $\kern0.37mm
u\circ x\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.15mm A\setminus N\kern0.37mm)\subseteq\{\kern0.37mm 0\kern0.37mm\}\,\} $ \newfline
$\impss03 \Cal V\not=\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm$ and $\kern0.37mm F=
\raise1.8mm\hbox{\font\≈=cmssi5\≈pr}\kern-.3mm
\LLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi{M\kern0.37mm\aars N_0} \,$, \KP9
\item \ $\mLrs03^p(\kern0.37mm\mu\kern0.15mm) =
\mvLrs03^p(\kern0.37mm\mu\,,\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) \KP1 $,
(6) \ $\mLrs03^p(\kern0.37mm\mu\kern0.15mm)\lfbb_C =
\mvLrs02^p(\,\mu\,,(\kern0.15mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm)\Reit1) \KP1 $, \KP{13} \inskipline0{-2}
(7) \ $\suptext{vc}0\Lrs03^p(\vcal Q\kern0.15mm) = \uniqset F:\eexi{\bosy K}\,\vcal Q\kern0.37mm$
is a quasi\kern0.37mm-\kern0.37mm\erm Euclidean \mathss37{\bosy K}--\,vector column \newskline{17}
and $\,\aall{\ell\,,\kern0.15mm\smb N\kern0.15mm,\kern0.15mm\mu\,,Q\kern0.37mm,\kern0.15mm\Yps\kern0.07mm,\kern0.07mm\vPi}\,
\vcal Q = (\kern0.15mm Q\kern0.37mm,\Yps\kern0.15mm,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm$ and $\kern0.37mm
\smb N \in \mathbb No \kern0.37mm$ and \newfline
$\ell\in\Lis(\,\Yps\Reit0\kern0.37mm,\tvbbR5^{\ssmb N}\kern0.37mm\big)\kern0.37mm$ and $\kern0.37mm
\mu = \seq{\KP1 r : A \subseteq Q \kern0.37mm$ and $\kern0.37mm
r = \Lebmef^{\kern0.37mm\ssmb N}\!\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\ell\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\,) \KP1 } $ \KP{8.95} \newfline
$\impss02 \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} Q \not = \hbox{\font\SweD =cmssbx10\SweD U}{} \kern0.37mm$ and $\kern0.37mm
F = \mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \KP1 $, \KP9 \inskipline0{-2}
(8) \ $\LLrs03^p(\kern0.15mm Q\,\sbi\Yps\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) =
\suptext{vc}0\Lrs03^p(\kern-0.3mm(\kern0.15mm Q\kern0.37mm,\Yps\kern0.15mm,\kern0.07mm\vPi\kern0.37mm)\kern-0.3mm) \KP1 $, \inskipline0{-2}
(9) \ $\LLrs03^p(\kern0.15mm Q\,\sbi\Yps) =
\LLrs03^p(\kern0.15mm Q\,\sbi\Yps\kern0.37mm,\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) \KP1 $,
(10) \ $\LLrs03^p(\kern0.15mm Q\,\sbi\Yps)\lfbb_C =
\LLrs03^p(\kern0.37mm Q\,\sbi\Yps\kern0.37mm,(\kern0.15mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm)\Reit1) \KP1 $, \KP{11.6} \inskipline0{-3.75}
(11) \ $\LLrs03^p(\kern0.15mm Q\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) = \uniqset F:\eexi\Yps\,\Yps\kern0.37mm$ is
quasi\kern0.37mm-\kern0.15mm usual over $\kern0.37mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm$ \newfline
and $\kern0.37mm Q\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\Yps\kern0.37mm$ and $\kern0.37mm
F = \LLrs03^p(\kern0.15mm Q\,\sbi\Yps\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) \KP1 $, \KP{17.4} \inskipline0{-3.75}
(12) \ $\LLrs03^p(\kern0.07mm Q\kern0.15mm) =
\LLrs03^p(\kern0.15mm Q\kern0.37mm,\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) \KP1 $,
(13) \ $\LLrs03^p(\kern0.07mm Q\kern0.15mm)\lfbb_C = \label{df $L^p(Q)_C$}
\LLrs03^p(\kern0.37mm Q\kern0.37mm,(\kern0.15mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm)\Reit1) \KP1 $. \KP{16.9}
\end{myLeftskip}\end{enumerate}
\end{constructions}
\begin{theorem}\label{L^p in TVS} | 3,505 | 362,273 | en |
train | 0.47.43 | and $\,\aall{\ell\,,\kern0.15mm\smb N\kern0.15mm,\kern0.15mm\mu\,,Q\kern0.37mm,\kern0.15mm\Yps\kern0.07mm,\kern0.07mm\vPi}\,
\vcal Q = (\kern0.15mm Q\kern0.37mm,\Yps\kern0.15mm,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm$ and $\kern0.37mm
\smb N \in \mathbb No \kern0.37mm$ and \newfline
$\ell\in\Lis(\,\Yps\Reit0\kern0.37mm,\tvbbR5^{\ssmb N}\kern0.37mm\big)\kern0.37mm$ and $\kern0.37mm
\mu = \seq{\KP1 r : A \subseteq Q \kern0.37mm$ and $\kern0.37mm
r = \Lebmef^{\kern0.37mm\ssmb N}\!\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\ell\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\,) \KP1 } $ \KP{8.95} \newfline
$\impss02 \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} Q \not = \hbox{\font\SweD =cmssbx10\SweD U}{} \kern0.37mm$ and $\kern0.37mm
F = \mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \KP1 $, \KP9 \inskipline0{-2}
(8) \ $\LLrs03^p(\kern0.15mm Q\,\sbi\Yps\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) =
\suptext{vc}0\Lrs03^p(\kern-0.3mm(\kern0.15mm Q\kern0.37mm,\Yps\kern0.15mm,\kern0.07mm\vPi\kern0.37mm)\kern-0.3mm) \KP1 $, \inskipline0{-2}
(9) \ $\LLrs03^p(\kern0.15mm Q\,\sbi\Yps) =
\LLrs03^p(\kern0.15mm Q\,\sbi\Yps\kern0.37mm,\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) \KP1 $,
(10) \ $\LLrs03^p(\kern0.15mm Q\,\sbi\Yps)\lfbb_C =
\LLrs03^p(\kern0.37mm Q\,\sbi\Yps\kern0.37mm,(\kern0.15mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm)\Reit1) \KP1 $, \KP{11.6} \inskipline0{-3.75}
(11) \ $\LLrs03^p(\kern0.15mm Q\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) = \uniqset F:\eexi\Yps\,\Yps\kern0.37mm$ is
quasi\kern0.37mm-\kern0.15mm usual over $\kern0.37mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm$ \newfline
and $\kern0.37mm Q\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\Yps\kern0.37mm$ and $\kern0.37mm
F = \LLrs03^p(\kern0.15mm Q\,\sbi\Yps\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) \KP1 $, \KP{17.4} \inskipline0{-3.75}
(12) \ $\LLrs03^p(\kern0.07mm Q\kern0.15mm) =
\LLrs03^p(\kern0.15mm Q\kern0.37mm,\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) \KP1 $,
(13) \ $\LLrs03^p(\kern0.07mm Q\kern0.15mm)\lfbb_C = \label{df $L^p(Q)_C$}
\LLrs03^p(\kern0.37mm Q\kern0.37mm,(\kern0.15mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm)\Reit1) \KP1 $. \KP{16.9}
\end{myLeftskip}\end{enumerate}
\end{constructions}
\begin{theorem}\label{L^p in TVS}
Let \ú$\,0\le p\le\lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure. With \ú$\,
\bosy K \in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\} $ also let \ú$\,\vPi\in\tvsps0(K)$ and either \ú$\, F =
\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) $ or \ú$ F =
\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \KPt8 $. Then \ú$\,F\in{\kern-0.63mm}$ $
\roman{TVS}\kern0.4mmps0(K)$ holds. If in addition \ú$\,1\le p$ and $\,\vPi$ is almost
suitable{\kern0.15mm\rm, }then \ú$\,F\in\roman{LCS}\kern0.4mmps0(K)$ holds with $\,F$ normable.
Furthermore{\kern0.15mm\rm, }for $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ any dominating norm for $\,\vPi$ it holds
that $\,\seqss43{\inf\kern0.15mm\big\{\KPt8\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu\kern-0.2mm
: x\in\smb X\,\} : \smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} $ is a compatible norm for $\,F\kern0.15mm$.
\end{theorem}
\begin{proof} Let \math{{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \math{ X =
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs} } and \vskip.5mm\centerline{$
M=
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm X\capss01\{\,x : (\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm \text{ is
finitely almost S measurable } \} $} \inskipline{.5}0
where S stands for either \q{simply} or \q{scalarly}. Then \math{X} is a
vector structure over \mathss32{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}, \,and it is a straightforward
standard exercise (\kern0.15mm to the reader\kern0.15mm) to verify that \math{M} is a vector
subspace in \mathss31{X}. So \math{X_{\kern0.37mm|\,M}} is a vector structure over \mathss32{
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}. Now for \math{\Cal V=
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi M} and \math{S = \kern0.37mm
\bigcap\kern0.37mm\big\{\KP{1.1}[\KP{1.2}\mathbb Z\KP{1.1}V\KP{1.1}]\vvs X\kern-0.3mm:
V\in\Cal V\KP1\} } we first see that \ú$\kern0.37mm S\subseteq{\kern-0.63mm}$ \linebreak
$M\kern0.37mm$ holds and
that \math{S} is a vector subspace in \mathss31{X}. Hence \math{X_{\kern0.37mm|\,S}}
is a vector structure over \mathss32{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}. For the set \math{
\Cal V\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 S} in \math{X_{\kern0.37mm|\,S}} one verifies that the properties
(\kern0.15mm\erm{NB}\,1\kern0.15mm) and (\kern0.15mm\erm{NB}\,2\kern0.15mm) given in \cite[p.\ 33]{Jr} hold.
Indeed, for given \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi} utilizing the short- \linebreak
hands \math{
\|\,x\,\|\subnu=\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu } and \mathss38{
\trN03\smb X\trNu2=\inf\kern0.15mm\big\{\KPt8\|\,x\,\|\subnu\kern-0.2mm:x\in\smb X\,\}
}, \,in the case \ú$\kern0.37mm p\not=0$ \linebreak
from Proposition \ref{Pro upint} on page \pageref{Pro upint}
above, putting \math{ \smb M = \smb A \kern-0.3mm \cdot
\sup\KPt8\{\,1\kern0.37mm,\kern0.07mm 2\KP1^{p^{-1}-\kern0.37mm 1\kern0.37mm}\big\} }
$\null
$ where \math{\smb A} is as on line 3 in (\ref{defi bqnor E}) on
page \pageref{defi bqnor E p} above, we first see \linebreak
that \math{
\|\KP1(\kern0.37mm x + y\kern0.37mm)\svs X\,\|\subnu \kern0.15mm \le \kern0.37mm
\smb M\KPt8\big(\kern0.37mm\|\,x\,\|\subnu + \|\,y\,\|\subnu\kern0.15mm) } holds for all \mathss31{
x\kern0.37mm,\kern0.15mm y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mms X}. This gives (\kern0.15mm\erm{NB}\,2\kern0.15mm) and
(\kern0.15mm\erm{NB}\,1\kern0.15mm) follows trivially from the property given on line 2 in
(\ref{defi bqnor E}) above. In the case \math{p=0} again (\kern0.15mm\erm{NB}\,1\kern0.15mm)
is trivial, and (\kern0.15mm\erm{NB}\,2\kern0.15mm) is seen by observing that we have
\math{1\le\smb A} and hence for all
\math{x\kern0.37mm,\kern0.15mm y\in S} and \math{\eta\in{}^{}\Cal Omega} it holds that \inskipline{.5}{19} | 3,748 | 362,273 | en |
train | 0.47.44 | where S stands for either \q{simply} or \q{scalarly}. Then \math{X} is a
vector structure over \mathss32{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}, \,and it is a straightforward
standard exercise (\kern0.15mm to the reader\kern0.15mm) to verify that \math{M} is a vector
subspace in \mathss31{X}. So \math{X_{\kern0.37mm|\,M}} is a vector structure over \mathss32{
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}. Now for \math{\Cal V=
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi M} and \math{S = \kern0.37mm
\bigcap\kern0.37mm\big\{\KP{1.1}[\KP{1.2}\mathbb Z\KP{1.1}V\KP{1.1}]\vvs X\kern-0.3mm:
V\in\Cal V\KP1\} } we first see that \ú$\kern0.37mm S\subseteq{\kern-0.63mm}$ \linebreak
$M\kern0.37mm$ holds and
that \math{S} is a vector subspace in \mathss31{X}. Hence \math{X_{\kern0.37mm|\,S}}
is a vector structure over \mathss32{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}. For the set \math{
\Cal V\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 S} in \math{X_{\kern0.37mm|\,S}} one verifies that the properties
(\kern0.15mm\erm{NB}\,1\kern0.15mm) and (\kern0.15mm\erm{NB}\,2\kern0.15mm) given in \cite[p.\ 33]{Jr} hold.
Indeed, for given \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi} utilizing the short- \linebreak
hands \math{
\|\,x\,\|\subnu=\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu } and \mathss38{
\trN03\smb X\trNu2=\inf\kern0.15mm\big\{\KPt8\|\,x\,\|\subnu\kern-0.2mm:x\in\smb X\,\}
}, \,in the case \ú$\kern0.37mm p\not=0$ \linebreak
from Proposition \ref{Pro upint} on page \pageref{Pro upint}
above, putting \math{ \smb M = \smb A \kern-0.3mm \cdot
\sup\KPt8\{\,1\kern0.37mm,\kern0.07mm 2\KP1^{p^{-1}-\kern0.37mm 1\kern0.37mm}\big\} }
$\null
$ where \math{\smb A} is as on line 3 in (\ref{defi bqnor E}) on
page \pageref{defi bqnor E p} above, we first see \linebreak
that \math{
\|\KP1(\kern0.37mm x + y\kern0.37mm)\svs X\,\|\subnu \kern0.15mm \le \kern0.37mm
\smb M\KPt8\big(\kern0.37mm\|\,x\,\|\subnu + \|\,y\,\|\subnu\kern0.15mm) } holds for all \mathss31{
x\kern0.37mm,\kern0.15mm y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mms X}. This gives (\kern0.15mm\erm{NB}\,2\kern0.15mm) and
(\kern0.15mm\erm{NB}\,1\kern0.15mm) follows trivially from the property given on line 2 in
(\ref{defi bqnor E}) above. In the case \math{p=0} again (\kern0.15mm\erm{NB}\,1\kern0.15mm)
is trivial, and (\kern0.15mm\erm{NB}\,2\kern0.15mm) is seen by observing that we have
\math{1\le\smb A} and hence for all
\math{x\kern0.37mm,\kern0.15mm y\in S} and \math{\eta\in{}^{}\Cal Omega} it holds that \inskipline{.5}{19}
$ \rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00(\kern0.37mm x + y\kern0.37mm)\vvs X\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
\le \rmmd\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\smb A\KPt8(\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta + \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)) $ \inskipline{.2}{53}
${} \le \smb A\KPt8(\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta +
\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) $ \inskipline{.5}0
whence further $\,\upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00
(\kern0.37mm x + y\kern0.37mm)\vvs X\,|\KP1 A\rmdss41\mu$ \inskipline{.4}{25}
${} \le \smb A\,\big(\kern0.37mm
\upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1|\KP1 A\rmdss41\mu\kern0.15mm + \kern-0.3mm
\upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 y\KP1|\KP1 A\rmdss41\mu\kern0.37mm) \KP1 $. \inskipline{.5}0
Consequently, we see that there is a unique vector topology \math{
\scrmt T\aR 1} for \math{X_{\kern0.37mm|\,S}} such that with \math{E =
(\kern0.15mm X_{\kern0.37mm|\,S}\kern0.37mm,\kern0.07mm\scrmt T\aR 1)} we have \math{\Cal V\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 S} a
filter base for \mathss35{\neiBoo E}.
Now letting \math{N\aar 0} be as on lines 6--7 in
Constructions \ref{defi $L^p$}\,(\ref{simpL^p}) or (\ref{ctr mvL_s^p})\,, it
is a simple matter to verify that \math{N\aar 0} is a vector subspace in \mathss37{
X_{\kern0.37mm|\,S}}. So for \math{Y= X_{\kern0.37mm|\,S}\,/\vsquotient N\aar 0} and \linebreak
\ú$F\aar 1=E\,/\tvsquotient N\aar 0\kern0.37mm$ we have \math{F\aar 1} a topological
vector space over \math{\bosy K} with \ú$\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\aar 1=Y\kern0.07mmp$. Let- ting \math{
\scrmt V\aar 0} and \math{\scrmt T} be as on lines 5--6 in
Constructions \ref{defi $L^p$}\,(\ref{preL^p_{MN_0}})\,, we have \math{ F =
(\kern0.37mm Y\kern0.07mmp,\kern0.07mm\scrmt T\,)} and from Lemma \ref{Le qtvs} on
page \pageref{Le qtvs} above we see that \math{\scrmt V\aar 0} is a filter
base for \mathss34{\neiBoo F\aar 1}, \,and hence \math{ F = F\aar 1 \in
\tvsps0(K) } holds.
To prove that \math{F\in\roman{TVS}\kern0.4mmps0(K)} holds, we need to show that \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}
is a Hausdorff topology. For this,
arbitrarily fixing \mathss38{\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.37mm\setminus\{\,\Bnull_F\}
}, \,in the case \math{p\not=0} it
suffices to show existence of some \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi} such that \math{
\trN03\smb X\trNu2\not=0 } holds.
To proceed, fixing any \mathss30{x\ar 0\in\smb X}, \,there are some
\math{u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } and
\math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ }
with $\int_{\,A}\kern0.37mm u\circss00 x\ar 0\rmdss01\mu\not=0$ and
hence also
$\int_{\,A\,}|\KP1 u\circss00 x\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\rmdss11\mu\not=0$ .
Consequently for \math{A\kern0.15mm\ar 1=
A\capss31\{\,\eta:
|\KP1 u\circss00 x\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1| \not= 0 \KP1\} }
and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.15mm=\kern0.37mm
\big\langle\KP1|\KP1 u\fvalss02\xi\KP1|:\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\KP1\rangle }
now \math{\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.15mm\ar 1 > 0} and
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Cal S_{_N}\kern0.15mm\vPi\subseteq\Bqnorm\vPi } hold. | 3,972 | 362,273 | en |
train | 0.47.45 | Now letting \math{N\aar 0} be as on lines 6--7 in
Constructions \ref{defi $L^p$}\,(\ref{simpL^p}) or (\ref{ctr mvL_s^p})\,, it
is a simple matter to verify that \math{N\aar 0} is a vector subspace in \mathss37{
X_{\kern0.37mm|\,S}}. So for \math{Y= X_{\kern0.37mm|\,S}\,/\vsquotient N\aar 0} and \linebreak
\ú$F\aar 1=E\,/\tvsquotient N\aar 0\kern0.37mm$ we have \math{F\aar 1} a topological
vector space over \math{\bosy K} with \ú$\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\aar 1=Y\kern0.07mmp$. Let- ting \math{
\scrmt V\aar 0} and \math{\scrmt T} be as on lines 5--6 in
Constructions \ref{defi $L^p$}\,(\ref{preL^p_{MN_0}})\,, we have \math{ F =
(\kern0.37mm Y\kern0.07mmp,\kern0.07mm\scrmt T\,)} and from Lemma \ref{Le qtvs} on
page \pageref{Le qtvs} above we see that \math{\scrmt V\aar 0} is a filter
base for \mathss34{\neiBoo F\aar 1}, \,and hence \math{ F = F\aar 1 \in
\tvsps0(K) } holds.
To prove that \math{F\in\roman{TVS}\kern0.4mmps0(K)} holds, we need to show that \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}
is a Hausdorff topology. For this,
arbitrarily fixing \mathss38{\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.37mm\setminus\{\,\Bnull_F\}
}, \,in the case \math{p\not=0} it
suffices to show existence of some \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi} such that \math{
\trN03\smb X\trNu2\not=0 } holds.
To proceed, fixing any \mathss30{x\ar 0\in\smb X}, \,there are some
\math{u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } and
\math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ }
with $\int_{\,A}\kern0.37mm u\circss00 x\ar 0\rmdss01\mu\not=0$ and
hence also
$\int_{\,A\,}|\KP1 u\circss00 x\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\rmdss11\mu\not=0$ .
Consequently for \math{A\kern0.15mm\ar 1=
A\capss31\{\,\eta:
|\KP1 u\circss00 x\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1| \not= 0 \KP1\} }
and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.15mm=\kern0.37mm
\big\langle\KP1|\KP1 u\fvalss02\xi\KP1|:\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\KP1\rangle }
now \math{\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.15mm\ar 1 > 0} and
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Cal S_{_N}\kern0.15mm\vPi\subseteq\Bqnorm\vPi } hold.
For every \math{x\in\smb X} and
\math{B\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+} we have
$\int_{\KPt8 B\,}u\circss00 x\rmdss11\mu
=\int_{\KPt8 B\,}u\circss00 x\ar 0\rmdss01\mu$
and hence there is some $N
\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
$u\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
=u\circss00 x\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$ and hence also
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$
holds for all $\eta\in A\kern0.15mm\ar 1\kern-0.63mm\setminus N$ .
In the case
\math{p\not=0} we hence
get \math{
0<\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\ar 0\,|\KP1 A\kern0.15mm\ar 1\,\|\Lnorss33^p_\mu
=\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1|\KP1 A\kern0.15mm\ar 1\,\|\Lnorss33^p_\mu
\le\|\,x\,\|\subnu } .
Since this holds for arbitrarily
given \math{x\in\smb X} we
consequently
obtain
\mathss38{0<\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\ar 0\,|\KP1 A\kern0.15mm\ar 1\,\|\Lnorss33^p_\mu
\le\inf\kern0.15mm\big\{\KPt8\|\,x\,\|\subnu\kern-0.2mm:x\in\smb X\,\}=
\trN03\smb X\trNu2 }.
In the case \math{p=0} the above deduction gives \vskip.5mm\centerline{$
0 < \int_{\,\aars A_1\kern-0.3mm}\rmmd\circ\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\ar 0\rmdss01\mu
= \int_{\,\aars A_1\kern-0.3mm}\rmmd\circ\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\rmdss11\mu
= \upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1|\KP1 A\kern0.15mm\ar 1\rmdss01\mu $} \inskipline{.3}0
for all \math{x\in\smb X} and hence taking \math{ \varepsilon =
\int_{\,\aars A_1\kern-0.3mm}\rmmd\circ\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\ar 0\rmdss01\mu } and \inskipline{.2}{7.3}
$V\aar 1 = M\capss21\{\,x :
\upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1|\KP1 A\kern0.15mm\ar 1\rmdss01\mu
< \varepsilon \KPt8 \} $ \inskipline{.4}0
and \math{V\aar 0 = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,\smb X : \smb X\capss02 V\aar 1
\not= \emptyset \KP1 \} } we have \mathss36{ \smb X \kern-0.3mm\not\in\kern0.15mm V\aar 0
\in \scrmt V\aar 0 }. \vskip.3mm
Finally assuming that also \math{1\le p} holds and that \math{\vPi} is almost
suitable, we fix any dominating norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \mathss31{\vPi}. Then by
Lemma \ref{Le suit dom} on page \pageref{Le suit dom} above, we see that the
set \math{ \{\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,\smb X\kern-0.3mm:n\KPt8\trN03\smb X\trNu2 < 1\KPt8
\} : n\in\rbb Z^+\kern0.37mm\big\} } a filter base for \mathss31{\neiBoo F}.
Consequently $\kern0.37mm F$ \linebreak
is locally convex and normable with a compatible norm as
asserted. Note that we get the triangle inequality \math{
\trN04(\kern0.37mm\smb X + \smb Y\,)\vvs Y\trNu2
\le \trN03\smb X\trNu2 + \trN03\smb Y\trNu4 } for \math{\smb X\kern0.15mm,\kern0.15mm\smb Y\kern0.15mm
\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F } from \vskip.4mm\centerline{$
\inf\,\{\,\|\,z\,\|\subnu\kern-0.2mm:z\in(\kern0.37mm\smb X+\smb Y\,)\vvs Y\,\}
\le \inf\,\{\,\|\,x\,\|\subnu\kern-0.2mm:x\in\smb X\,\} +
\inf\,\{\,\|\,y\,\|\subnu\kern-0.2mm:y\in\smb Y\KP1\} \KP1 $,} \inskipline{.4}0
and that the implication \math{\trN03\smb X\trNu2=0\impss33 \smb X=\Bnull_F}
holds for all \math{\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} since we already know that \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F} is a Hausdorff topology.
\end{proof} | 3,731 | 362,273 | en |
train | 0.47.46 | \begin{lemma}\label{Le 0_{L^p}}
Let \ú$\,0 \le p \le\lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure. With \ú$\,
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ also let \ú$\,\vPi\in\roman{LCS}\kern0.4mmps0(K)$ be normable{\kern0.15mm\rm, }and
let \ú$\,F=\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ and \ú$\,x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F$
and \ú$\,y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.15mm$. Then \ú$\,y\in\smb X$ holds if and only
if for every $\,A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ there is some $\,
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ with $\,
x\KP1|\KP1(\kern0.15mm A\kern0.07mm\setminus N\kern0.37mm)\subseteq y \, $.
\end{lemma}
\begin{proof} The asserted sufficiency being trivial, we only verify
necessity. So letting \math{y\in\smb X} and \math{A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } we need to get some \math{N\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } with \mathss34{
x\KP1|\KP1(\kern0.15mm A\kern0.07mm\setminus N\kern0.37mm)\subseteq y }. Now we first find some \math{
N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and simple sequences
\math{\bosy\sigma\aR 1} and \math{\bosy\sigma\aR 2} in
\newline
\math{(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.15mm A\kern0.07mm\setminus N\aar 1)\,,\kern0.07mm\vPi\kern0.37mm) }
with \math{
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy\sigma\aR 1\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } and \math{
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy\sigma\aR 2\to y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \mathss34{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} for all \math{\eta\in A\kern0.07mm\setminus N\aar 1}. Then
letting \math{S} be the linear
\mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}--\,span of \math{
\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}(\kern0.37mm\bosy\sigma\aR 1\kern-0.2mm\cup\kern0.37mm\bosy\sigma\aR 2\kern0.07mm) } we
have
\math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 S } a separable topology, and for \math{B\ar 1}
the closed unit dual ball corresponding to some fixed compatible norm for
\math{\vPi} and for \math{\scrmt T=
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.07mm\vPi_{\kern0.37mm/\,S}\kern0.07mm)\dlsigss00\kern0.07mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 B\ar 1 } hence
by \cite[Proposition 8.5.3\kern0.37mm, p.\ 157]{Jr} we see that
\math{\scrmt T} is a
metrizable and separable topology. Let then
\math{D} be countable and \mathss37{\scrmt T}--\,dense. Now
by {\sl Hahn\,--\,Banach\kern0.15mm} for
every fixed \math{u\in D} and for all \math{
B\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.15mm A\kern0.07mm\setminus N\aar 1) } we have \math{
\int_{\KP1 B\,}u\circss00 x\rmdss11\mu =
\int_{\KP1 B\,}u\circss00 y\rmdss11\mu } and hence there is some
\math{N\kern0.15mmrim1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } with \mathss34{
u\circss00 x\KP1|\KP1(\kern0.15mm A\kern0.07mm\setminus N\kern0.15mmrim1\kern0.15mm)\subseteq
u\circss00 y }. By {\sl countable choice\kern0.15mm} taking the union of these
\math{N\kern0.15mmrim1} for
\math{u\in D} we obtain \math{N} with
\math{N\aar 1\subseteq N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and \mathss34{
u\circss00 x\KP1|\KP1(\kern0.15mm A\kern0.07mm\setminus N\kern0.37mm)\subseteq
u\circss00 y } for all \math{u\in D}. Then to get \math{
x\KP1|\KP1(\kern0.15mm A\kern0.07mm\setminus N\kern0.37mm)\subseteq y } arbitrarily fixing \math{\eta\in
A\kern0.07mm\setminus N } and \math{v\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) }
by {\sl Hahn\,--\,Banach\kern0.15mm} it suffices to have
\mathss34{v\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=v\circss00 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta}. Now we find some
\math{\bosy u\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,D } with
\math{\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm\bosy u \to v\fvalss01\xi }
for all
\newline
\math{\xi\in\{\,x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm,\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KPt8\} } and then we get
\inskipline{.2}{21}
$ v\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
= \lim\,(\kern0.37mm\roman{ev}\kern0.15mm\sbi{\emath x\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm}\eta}\kern-0.2mm\circ\kern0.15mm\bosy u\kern0.37mm)
= \lim\,(\kern0.37mm\roman{ev}\kern0.15mm\sbi{y\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm}\eta}\kern-0.2mm\circ\kern0.15mm\bosy u\kern0.37mm)
= v\circss00 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta \, $.
\end{proof} | 3,038 | 362,273 | en |
train | 0.47.47 | From Lemma \ref{Le 0_{L^p}} we see in particular that in the case where \math{
\mu} is \rsigma5finite, elements \mathss03{x\kern0.37mm,\kern0.15mm y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}
represent the same vector of \math{F} if and only if they are equal almost
everywhere in the classical sense. In the case \math{p\not=0} even without
\rsigma5finiteness we also see that corresponding to any given compatible norm \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \math{\vPi} we have the equality \math{
\inf\kern0.15mm\big\{\KPt8\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 z\KP1\|\Lnorss33^p_\mu \kern-0.2mm : \label{discus inf N = N}
z\in\smb X\,\} = \|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu } for \mathss31{
x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}. \vskip.3mm
On page \pageref{triv L^p exa} above we noted that for \math{ 0 \le p \le
\lower1.05mm\hbox{$^+$}\infty } and e.g.\ for \ú$\kern0.37mm F=\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) $ \linebreak
with \math{\vPi=\LLrs42^{\frac 12}(\ssbb44 I) } we have \math{F} trivial in
the sense that \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F=\{\,\Bnull_F\} } holds. However, in
Constructions \ref{defi $L^p$}\,(\ref{preL^p_{MN_0}}) taking \math{ M =
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F=\Bnull_F} and \vskip.5mm\centerline{$
N\aar 0 = M\capss21\{\,x:\aall{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi}\,
\upint\kern0.37mm \Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\rmdss41\mu = 0\KP1\} $} \inskipline{.5}0
we generally get a nontrivial space \math{ E =
\raise1.7mm\hbox{\font\≈=cmssi5\≈pr}\kern-.3mm
\LLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.15mm\sbi{M\kern0.37mm\aars N_0} } such that
e.g.\ for \ú$\kern0.37mm p=\frac 12$ \linebreak
and \math{\mu=\Lebmef^{}\,|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\bbI } the
spaces \math{E} and \math{\LLrs03^p(\ssbb40 I\times\ssbb04 I) } become
naturally linearly homeomorphic. We leave the proof as an exercise to the
reader.
\begin{theorem}\label{Th L_s^p Ba}
Let \ú$\,1\le p\le\lower1.05mm\hbox{$^+$}\infty$ and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in
\roman{LCS}\kern0.4mmps0(K)$ be suitable. Let $\,\mu$ be a positive measure such that in the
case \ú$\,p=\lower1.05mm\hbox{$^+$}\infty$ it holds that $\,\mu$ is almost decomposable. Then $\,
\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \in \roman{BaS}\kern0.4mmps0(K) $ holds. If in addition $\,
\vPi$ is \eit Ba- nachable{\kern0.15mm\rm, }then also $\,
\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \in \roman{BaS}\kern0.4mmps0(K) $ holds.
\end{theorem}
\begin{proof} We give the proof for \math{\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) }
and leave it as an exercise to the reader to make the slight modifications
that are needed to get the assertion related to \math{
\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } that is classical in the cases where \math{
p\not=\lower1.05mm\hbox{$^+$}\infty} holds or \math{\mu} is \rsigma5finite. For hint we only
mention that \cite[Theorem 4.2.2\kern0.37mm, p.\ 95]{Du} and
\cite[Corollary 4.2.7, p.\ 97]{Du} together can be utilized to deduce that for
the obtained \math{y} it then holds that \mathss03{
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is finitely almost simply measurable.
Now we put \math{{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \math{ X =
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.37mm]_{vs} } and \mathss38{ F =
\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)}, \,and let \linebreak
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm$ be any dominating
norm for \mathss31{\vPi}. By Theorem \ref{L^p in TVS} only completeness of \math{
F} has to be verified. For this, it suffices to show that
for any \PouN$\sevib X \in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F$ with
\PouN$\trN05\sevib X\fvalss32 i\trNu6 <
4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)}$ for all $i\in\mathbb No\,$, the sequence
\math{\sevib Y=
\seqss33{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm F\text{\,-\kern0.15mm}\sum_{\,k\kern0.37mm\in\kern0.37mm i^+\kern0.37mm}(\kern0.15mm
\sevib X\fvalss32 k\kern0.37mm):i\in\mathbb No} } converges in the
topology $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F$. In order to get this, we first take some $\bosy x \in
\prod{_{_{\kern-.3mm\bold c\kern.15mm}}}\kern0.15mm\sevib X$ with
$\|\KP1\bosy x\fvalss12 i\KP1\|\subnu <
4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)}$ for all $i\in\mathbb No\,$,
and put $\bosy y=
\seqss33{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm\vPi\text{\,-\kern0.15mm}\sum_{\,k\kern0.37mm\in\kern0.37mm i^+\kern0.37mm}(\kern0.37mm
\bosy x\fvalss12 k\kern0.37mm):i\in\mathbb No}\,$.
First considering the case \mathss35{p=\lower1.05mm\hbox{$^+$}\infty}, \,letting \math{\scrmt A}
and \math{N\kern0.15mmrim1} be as in Definitions \ref{df decomp}\,(2) on page \pageref{decos A}
above, let \math{\scrmt N\ar 1} be the set of all \math{(\kern0.15mm A\,,\kern0.07mm N\aar 1) }
with \math{A\in\scrmt A} and \ú$\kern0.37mm N\aar 1 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} $ \linebreak
and such that \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ(\kern0.37mm\bosy x\fvalss01 i\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
< 4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)} } holds for all \math{i\in\mathbb No} and \math{
\eta\in A\kern0.15mm\setminus N\aar 1}. Then we have \mathss32{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt N\ar 1}, \,and hence by the {\sl axiom of choice\kern0.15mm} there is a
function \ú$\kern0.37mm\scrmt N\subseteq\scrmt N\ar 1$ \linebreak
with \mathss35{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt N}. Now taking \mathss35{N\kern0.15mmrimm1 =
N\kern0.15mmrim1\cupss24\bigcup\,{}^{}{\rm rng}\,{}_{{}^{}}\scrmt N }, \,we see that \math{N\kern0.15mmrimm1} is \mathss37{
\mu}--\,negli- gible and such that \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ(\kern0.37mm\bosy x\fvalss01 i\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
< 4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)} } holds for \math{i\in\mathbb No} and \mathss32{
\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.15mmrimm1}. | 3,872 | 362,273 | en |
train | 0.47.48 | First considering the case \mathss35{p=\lower1.05mm\hbox{$^+$}\infty}, \,letting \math{\scrmt A}
and \math{N\kern0.15mmrim1} be as in Definitions \ref{df decomp}\,(2) on page \pageref{decos A}
above, let \math{\scrmt N\ar 1} be the set of all \math{(\kern0.15mm A\,,\kern0.07mm N\aar 1) }
with \math{A\in\scrmt A} and \ú$\kern0.37mm N\aar 1 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} $ \linebreak
and such that \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ(\kern0.37mm\bosy x\fvalss01 i\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
< 4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)} } holds for all \math{i\in\mathbb No} and \math{
\eta\in A\kern0.15mm\setminus N\aar 1}. Then we have \mathss32{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt N\ar 1}, \,and hence by the {\sl axiom of choice\kern0.15mm} there is a
function \ú$\kern0.37mm\scrmt N\subseteq\scrmt N\ar 1$ \linebreak
with \mathss35{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt N}. Now taking \mathss35{N\kern0.15mmrimm1 =
N\kern0.15mmrim1\cupss24\bigcup\,{}^{}{\rm rng}\,{}_{{}^{}}\scrmt N }, \,we see that \math{N\kern0.15mmrimm1} is \mathss37{
\mu}--\,negli- gible and such that \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ(\kern0.37mm\bosy x\fvalss01 i\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
< 4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)} } holds for \math{i\in\mathbb No} and \mathss32{
\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.15mmrimm1}.
Letting \math{\vPi\ar 0} be the Banach{\sl able\kern0.15mm} space determined by the
norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \mathss30{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}, \,from the above we see that for
every fixed \math{\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.15mmrimm1 } the sequence \math{
\roman{ev}\sbi\eta\circ\kern0.07mm\bosy y } converges in the topology \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\aar 0} and hence also in the weaker topology \mathss30{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}.
Taking \vskip.3mm\centerline{$
y = N\kern0.15mmrimm1\kern-0.3mm\times\kern-0.2mm\{\,\Bnull_\vPi\} \cupss22 \{ \,
(\kern0.37mm\eta\kern0.37mm,\kern0.07mm\xi\kern0.37mm):\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.15mmrimm1\kern0.15mm\text{ and }\kern0.37mm
\roman{ev}\sbi\eta\circ\kern0.07mm\bosy y\to\xi\kern0.37mm\text{ in top }\kern0.37mm
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\aar 0\,\} \KP1 $,} \inskipline{.3}0
by Lemma \ref{Le deco meas} on page \pageref{Le deco meas} above \math{
(\KPt5 y\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is finitely almost scalarly measurable. It
is also a simple exercise to see that \math{y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} holds, and
that for the unique class \math{\smb Y} with \math{y\in\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} we
indeed have \math{\ebit Y\to\smb Y} in top \mathss30{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}.
Next, for the case \mathss35{p<\lower1.05mm\hbox{$^+$}\infty}, \,we choose a sequence \math{
\bosy u} of fully positive \mathss37{\mu}--\,mea- surable functions such that
for all \math{i\in\mathbb No} and \math{\eta\in{}^{}\Cal Omega} we have
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ(\kern0.37mm\bosy x\fvalss01 i\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\eta\le
\bosy u\fvalss01 i\fvalss10\eta$ and $
\int_{\KP1{}^{}\Cal Omega}\kern0.37mm\aabs99^p\circss01(\kern0.37mm
\bosy u\fvalss01 i\kern0.37mm)\rmdss11\mu
< 4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)\,p}\,$. Putting
$\roman A\,i=\{\,\eta:
\bosy u\fvalss01 i\fvalss10\eta
\ge 2^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)}\,\big\}$ and
$\roman B\,i=
\bigcup\,\{\,\roman A\,j:i\subseteq j\in\mathbb No\,\}\,$, for $i\in\mathbb No$ we
have $
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\roman A\,i\cdot 2^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)\,p}
\le\int_{\KP1\roman A\kern0.37mm i}\kern0.37mm\aabs99^p\circss01(\kern0.37mm
\bosy u\fvalss01 i\kern0.37mm)\rmdss11\mu
\le\int_{\KP1{}^{}\Cal Omega}\kern0.37mm\aabs99^p\circss01(\kern0.37mm
\bosy u\fvalss01 i\kern0.37mm)\rmdss11\mu
< 4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)\,p}\,$
and hence $
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\roman A\,i < 2^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)\,p}\,$, whence
further $
\mu\fvalss13\roman B\,i < 2\KP1^{(\kern0.15mm 1\kern0.37mm-\kern0.37mm(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm))\,p}\,$,
and
consequently for
$N=\bigcap\,\{\KP1\roman B\,i:i\in\mathbb No\,\}\,$, we get $
N\in\mu\invss34\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.15mm 0\kern0.15mm\}\,$. For each fixed $\eta\in
{}^{}\Cal Omega\kern0.07mm\setminus N$ there
is $i\ar 0\in\mathbb No$ with $\eta\not\in
\roman A\,i$ for all $i\in\mathbb No\kern-0.3mm\setminus\kern0.07mm i\ar 0\,$. Hence for $i\in
\mathbb No\kern-0.3mm\setminus\kern0.07mm i\ar 0\,$, we have
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ(\kern0.37mm\bosy x\fvalss01 i\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\eta\le
\bosy u\fvalss01 i\fvalss10\eta <
2^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)}\kern0.15mm$, and consequently the sequence $
\,\roman{ev}\sbi\eta\circ\kern0.07mm\bosy y$
converges in the topology $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\aar 0\,$. It follows that
there is a function $y:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi$ with $y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=\Bnull_\vPi$ for $
\eta\in N$ and
$\roman{ev}\sbi\eta\circ\kern0.07mm\bosy y\to y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$ in top
$\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ for $\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.15mm$. This immediately
gives that
$(\kern0.37mm u\circss01 y\,;\kern0.07mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ is finitely
measurable for every $u\in\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)\,$. | 4,017 | 362,273 | en |
train | 0.47.49 | and hence $
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\roman A\,i < 2^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)\,p}\,$, whence
further $
\mu\fvalss13\roman B\,i < 2\KP1^{(\kern0.15mm 1\kern0.37mm-\kern0.37mm(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm))\,p}\,$,
and
consequently for
$N=\bigcap\,\{\KP1\roman B\,i:i\in\mathbb No\,\}\,$, we get $
N\in\mu\invss34\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.15mm 0\kern0.15mm\}\,$. For each fixed $\eta\in
{}^{}\Cal Omega\kern0.07mm\setminus N$ there
is $i\ar 0\in\mathbb No$ with $\eta\not\in
\roman A\,i$ for all $i\in\mathbb No\kern-0.3mm\setminus\kern0.07mm i\ar 0\,$. Hence for $i\in
\mathbb No\kern-0.3mm\setminus\kern0.07mm i\ar 0\,$, we have
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ(\kern0.37mm\bosy x\fvalss01 i\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\eta\le
\bosy u\fvalss01 i\fvalss10\eta <
2^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)}\kern0.15mm$, and consequently the sequence $
\,\roman{ev}\sbi\eta\circ\kern0.07mm\bosy y$
converges in the topology $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\aar 0\,$. It follows that
there is a function $y:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi$ with $y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=\Bnull_\vPi$ for $
\eta\in N$ and
$\roman{ev}\sbi\eta\circ\kern0.07mm\bosy y\to y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$ in top
$\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ for $\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.15mm$. This immediately
gives that
$(\kern0.37mm u\circss01 y\,;\kern0.07mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ is finitely
measurable for every $u\in\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)\,$.
To show that $y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.15mm$, we must verify that
$\upint\kern0.37mm\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 y\rmdss11\mu < \lower1.05mm\hbox{$^+$}\infty$
holds.
For each fixed $\eta\in
{}^{}\Cal Omega\kern0.07mm\setminus N$ we have
\vskip1mm
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\text{\KP1-}\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy x\KP1|\KP1 i\kern0.37mm))) $ \inskipline{.7}{15.5}
${}
= \liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\text{\KP1-}\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy x\KP1|\KP1 i\kern0.37mm)))$ \inskipline{.7}{15.5}
${}
\le \liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\sum\KP1(\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ\kern0.15mm\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy x\KP1|\KP1 i\kern0.37mm)
\le \liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy u\KP1|\KP1 i\kern0.37mm)$
\vskip1mm
\noindent
and hence by Fatou's lemma we get \vskip1mm
$ \upint\kern0.37mm\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 y\rmdss11\mu
\le
\int_{\KP{1.1}{}^{}\Cal Omega\,}(\,
\liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy u\KP1|\KP1 i\kern0.37mm)
\sbig)3\RHB{.3}{^{\,p}}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline{.7}{30.2}
${}=\int_{\KP{1.1}{}^{}\Cal Omega\,}
\liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\big(\kern0.15mm\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy u\KP1|\KP1 i\kern0.37mm)
\sbig)3\RHB{.3}{^{\,p}}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline{.7}{30.2}
${}\le\liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty}\int_{\KP{1.1}{}^{}\Cal Omega\kern0.15mm}
\big(\kern0.15mm\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy u\KP1|\KP1 i\kern0.37mm)
\sbig)3\RHB{.3}{^{\,p}}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline{.7}{30.2}
${}\le
\liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\big(\kern0.15mm
\sum_{\,k\kern0.37mm\in\kern0.37mm i\kern0.15mm}
\big(\kern0.15mm\int_{\KP{1.1}{}^{}\Cal Omega\,}
(\kern0.37mm\bosy u\fvalss01 k\fvalss10\eta\kern0.37mm)\RHB{.3}{\KP1^p}
\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm))\KP1^{p^{-1}}\big)
\RHB{.7}{\,^p}$ \inskipline{.7}{30.2}
${} \le \liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\big(\kern0.15mm
\sum_{\,k\kern0.37mm\in\kern0.37mm i\,}4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm k\kern0.07mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm)}\big)
\RHB{.7}{\,^p}
= \big(\kern0.15mm\frac43\kern0.15mm\big)\RHB{.7}{\,^p}<\lower1.05mm\hbox{$^+$}\infty$ . \inskipline10
So we have $y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.15mm$, and hence there is
$\smb Y$ with $y\in\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F$. It remains to show that
$\sevib Y\to\smb Y$ in top $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F$. For this, similarly as above, we
compute \vskip1mm
$ \upint\kern0.37mm\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10
(\kern0.37mm\bosy y\fvalss01 i - y\kern0.37mm)\vvs X\rmdss11\mu
\le \int_{\KP1{}^{}\Cal Omega}\,(\,\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}\bosy u\fvalss01 k\fvalss10\eta
\,)\RHB{.3}{\KPt8^p}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm) $ \inskipline1{30}
${}=\int_{\KP1{}^{}\Cal Omega}\kern0.37mm\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\big(\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}\bosy u\fvalss01 k\fvalss10\eta\,)
\RHB{.3}{\KPt8^p}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline1{15} | 3,997 | 362,273 | en |
train | 0.47.50 | ${}=\int_{\KP{1.1}{}^{}\Cal Omega\,}
\liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\big(\kern0.15mm\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy u\KP1|\KP1 i\kern0.37mm)
\sbig)3\RHB{.3}{^{\,p}}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline{.7}{30.2}
${}\le\liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty}\int_{\KP{1.1}{}^{}\Cal Omega\kern0.15mm}
\big(\kern0.15mm\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy u\KP1|\KP1 i\kern0.37mm)
\sbig)3\RHB{.3}{^{\,p}}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline{.7}{30.2}
${}\le
\liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\big(\kern0.15mm
\sum_{\,k\kern0.37mm\in\kern0.37mm i\kern0.15mm}
\big(\kern0.15mm\int_{\KP{1.1}{}^{}\Cal Omega\,}
(\kern0.37mm\bosy u\fvalss01 k\fvalss10\eta\kern0.37mm)\RHB{.3}{\KP1^p}
\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm))\KP1^{p^{-1}}\big)
\RHB{.7}{\,^p}$ \inskipline{.7}{30.2}
${} \le \liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\big(\kern0.15mm
\sum_{\,k\kern0.37mm\in\kern0.37mm i\,}4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm k\kern0.07mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm)}\big)
\RHB{.7}{\,^p}
= \big(\kern0.15mm\frac43\kern0.15mm\big)\RHB{.7}{\,^p}<\lower1.05mm\hbox{$^+$}\infty$ . \inskipline10
So we have $y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.15mm$, and hence there is
$\smb Y$ with $y\in\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F$. It remains to show that
$\sevib Y\to\smb Y$ in top $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F$. For this, similarly as above, we
compute \vskip1mm
$ \upint\kern0.37mm\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10
(\kern0.37mm\bosy y\fvalss01 i - y\kern0.37mm)\vvs X\rmdss11\mu
\le \int_{\KP1{}^{}\Cal Omega}\,(\,\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}\bosy u\fvalss01 k\fvalss10\eta
\,)\RHB{.3}{\KPt8^p}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm) $ \inskipline1{30}
${}=\int_{\KP1{}^{}\Cal Omega}\kern0.37mm\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\big(\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}\bosy u\fvalss01 k\fvalss10\eta\,)
\RHB{.3}{\KPt8^p}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline1{15}
${}\le\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\int_{\KP1{}^{}\Cal Omega}\kern0.37mm\big(\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}
\bosy u\fvalss01 k\fvalss10\eta\,)\RHB{.3}{\KPt8^p}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline1{15}
${}\le\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\big(\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+}\big(\int_{\KP1{}^{}\Cal Omega}\kern0.37mm(\kern0.37mm
\bosy u\fvalss01 k\fvalss10\eta\,)\RHB{.3}{\KPt8^p}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
)\RHB{.3}{\KPt8^{p^{-1}}}\kern0.15mm\big)^{\,p}$ \inskipline1{15}
${}\le\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\big(\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}
4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm k\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm)}\kern0.15mm\big)\RHB{.7}{\,^p}
= \lim_{\,j\kern0.37mm\to\kern0.37mm\infty}\big(\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}
4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm k\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm)}\kern0.15mm\big)\RHB{.7}{\,^p}$ \inskipline1{15}
${}=(\kern0.37mm 3^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm)}\kern0.15mm\big)\RHB{.7}{\,^p}
\to 0 \,$ as $\, i\to\infty \KPt7 $, \,whence the assertion.
\end{proof} | 2,404 | 362,273 | en |
train | 0.47.51 | \begin{corollary}\label{Cor L^p Ban}
Let \ú$\,1\le p\le\lower1.05mm\hbox{$^+$}\infty$ and \ú$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ with \ú$\,\bosy K\in
\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}\KPt7$. Let $\,\mu$ be a positive measure such that in the case \ú$\,p=
\lower1.05mm\hbox{$^+$}\infty$ it holds that $\,\mu$ is decomposable. Also let \ú$\, F =
\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) ${\KP1\rm, }or let \ú$\, F =
\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with $\,\vPi$ reflexive or $\,
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ a separable topology. Then \ú$\,F\in\roman{BaS}\kern0.4mmps0(K)$ holds. Furthermore \ú$\,
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) =
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$ holds when the space $\,\vPi$ is
reflexive.
\end{corollary}
\begin{proof} The first alternative is immediate. For the second in the
separable case we note that by Proposition \ref{pro-mea-equ} on page \pageref{pro-mea-equ}
for \math{\vPi\aar 1=\vPi\dualsigma0} we have \mathss37{ F =
\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\aar 1\kern0.07mm) }. Since the conditions of Theorem \ref{Th L_s^p Ba}
for \math{\vPi\aar 1} in place of \math{\vPi} hold true, consequently the
assertion follows.
For the reflexive case putting \mathss37{ F\aar 0 =
\LLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) }, \,it suffices to verify that \math{
F=F\aar 0} holds, and this in turn follows if \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 0} can
be established. Trivially every \math{\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 0} is contained in
some \mathss30{\smb Y\aR 1\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}. For the converse, letting \mathss30{ y
\in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} and \mathss30{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+
}, \,there is \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that \math{
(\kern0.37mm y\KP1|\KP1 B\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } is
simply measurable for \mathss30{B=A\kern0.07mm\setminus N}. Hence by Proposition \ref{Pro rfx si mea}
on page \pageref{Pro rfx si mea} above also \mathss03{
(\kern0.37mm y\KP1|\KP1 B\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } is
simply measurable, and so \math{
(\kern0.37mm y\KP1|\KP1 A\KPt8;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } is
almost simply measurable. Having here \math{A} arbitrary, consequently \math{
y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 0} holds, and we are done.
\end{proof} | 1,634 | 362,273 | en |
train | 0.47.52 | Note that by Banach\,--\,Steinhaus for the second alternative in \label{non Ban but Bai}
Corollary \ref{Cor L^p Ban} we could weaken the assumption that \math{
\vPi\in\roman{BaS}\kern0.4mmps0(K)} hold to requiring \ú$\kern0.37mm\vPi\in\roman{LCS}\kern0.4mmps0(K)$ \linebreak
with \math{\vPi}
normable and barrelled. For an example of an incomplete normable barrelled
space, see e.g.\ \cite[5.7.\kern0.15mm\erm B\kern0.37mm, p.\ 97]{Jr}\,. \vskip3mm
Since in \cite[10.7, p.\ 214]{Jr} the term {\sl quasi\kern0.37mm-\kern0.15mm normable\kern0.15mm} is
reserved for a different meaning, for Proposition \ref{Pro simp Lp dense}
below we here agree to say that a real or complex topological vector space \math{
E} is {\it pseudonormable\kern0.37mm} if{}f there is some \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm E} with \mathss03{
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm{\big[\kern0.15mm} :
n\in\rbb Z^+\kern0.15mm\big\} } a filter base for \mathss34{\neiBoo E}.
Now the Hausdorff quotients of pseudonormable spaces correspond to the locally
bounded spaces in the following sense. If \math{E} is pseudonormable, then for \linebreak \mathss03{
F=E\,/\tvsquotient\kern-0.3mm\bigcap\KPt8\neiBoo E} we have that \math{F} is locally
bounded, and from \cite[Theorem 6.8.3\kern0.37mm, p.\ 114]{Jr} it follows existence
of \math{r\kern0.15mm,\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 0} with \math{0 < r \le 1} and \math{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 0 \in
\Cal S\sbi{\kern0.15mm\emath r\,}F } with \linebreak
\mathss03{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 0\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\subseteq\{\,\Bnull_F\} } and \math{
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 0\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm{\big[\kern0.15mm}
: n\in\rbb Z^+\kern0.15mm\big\} } a filter base for \mathss30{\neiBoo F}. Now with \math{
\tweq=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm\smb X\kern0.15mm) :
x\in\smb X\,\} } taking \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 0\circ\kern0.15mm\tweq} we see that \mathss30{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Cal S\sbi{\kern0.15mm\emath r\,}E } \linebreak
with also \math{
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm{\big[\kern0.15mm} :
n\in\rbb Z^+\kern0.15mm\big\} } a filter base for \mathss34{\neiBoo E}. Thus the
{\sl zero neighbourhoods of a pseudonormable space \math{E} are given by a
single continuous \mathss35{r}--\,seminorm\kern0.15mm} which is an \mathss35{r
}--\,norm if \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E} is a Hausdorff topogy. In particular, the
Hausdorff pseudonormable spaces are precisely the locally bounded ones.
\begin{proposition}\label{Pro simp Lp dense}
Let $\,p\in\rbb R^+$ and let $\,\mu$ be a positive measure{\kern0.15mm\rm, }and with
$\,\bosy K\in{}$
\newline
$\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let $\,\vPi\in\tvsps0(K)$ be pseudonormable. Also let
$\,F=\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ and
\newline
$\,D=
{\kern-0.63mm}$ $ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,\smb X:
\eexi{x\in\smb X}\,x\kern0.37mm\text{ is simple in }\kern0.37mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\KPt8\} \KPt8 $. Then $\,D$ is $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\,
$--\,dense.
\end{proposition}
\begin{proof} Put \math{{}^{}\Cal Omega=\kern0.15mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and let \mathss31{x\in
\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}. Let \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Cal S\sbi{\kern0.15mm\emath r\,}\vPi} be such
that \math{
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm{\big[\kern0.15mm} :
n\in\rbb Z^+\kern0.15mm\big\} } is a filter base for \mathss31{\neiBoo\vPi}. Then
we take
some fully positive
\mathss37{\mu}--\,measurable \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm} with \math{
\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\le\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm}
and \mathss36{
\int_{\KPp1.1{}^{}\Cal Omega\,} | 2,829 | 362,273 | en |
train | 0.47.53 | \begin{proposition}\label{Pro simp Lp dense}
Let $\,p\in\rbb R^+$ and let $\,\mu$ be a positive measure{\kern0.15mm\rm, }and with
$\,\bosy K\in{}$
\newline
$\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let $\,\vPi\in\tvsps0(K)$ be pseudonormable. Also let
$\,F=\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ and
\newline
$\,D=
{\kern-0.63mm}$ $ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,\smb X:
\eexi{x\in\smb X}\,x\kern0.37mm\text{ is simple in }\kern0.37mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\KPt8\} \KPt8 $. Then $\,D$ is $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\,
$--\,dense.
\end{proposition}
\begin{proof} Put \math{{}^{}\Cal Omega=\kern0.15mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and let \mathss31{x\in
\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}. Let \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Cal S\sbi{\kern0.15mm\emath r\,}\vPi} be such
that \math{
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm{\big[\kern0.15mm} :
n\in\rbb Z^+\kern0.15mm\big\} } is a filter base for \mathss31{\neiBoo\vPi}. Then
we take
some fully positive
\mathss37{\mu}--\,measurable \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm} with \math{
\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\le\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm}
and \mathss36{
\int_{\KPp1.1{}^{}\Cal Omega\,}
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu <
\lower1.05mm\hbox{$^+$}\infty}. Putting
\newline
\math{\roman A\,n=
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm[\KP1 2\KP1^{\emath n\kern0.37mm - \kern0.37mm 1}\kern0.07mm,\kern0.07mm
2\KPt9^{\emath n}\kern0.37mm{[\kern0.15mm} } we have \math{\{\,\roman A\,n:n\in\ssbb03 Z\,\}
\subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+}, \,and by
{\sl countable choice\kern0.15mm} we find \math{
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and
\math{
\bmii8 S\in\kern0.15mm^{\ssbb05 Z}\big(\KPt6^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm) } such that for
every \math{n\in\mathbb Z} we have \math{\bmii8 S\fvalss20 n} a
simple sequence in
\math{(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\roman A\,n\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) } with
\math{\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm(\kern0.15mm\bmii8 S\fvalss20 n\kern0.37mm)
\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}
for all \mathss30{\eta\in\roman A\,n\setminus N}. Then let
\math{\bmii8 S\ar 1\in\kern0.15mm^{\ssbb05 Z}\big(\KPt6^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm) } be
the unique one such that for all
\math{n\in\mathbb Z} and \math{i\in\mathbb No} and \math{
\sigma=\bmii8 S\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} n\fvalss01 i} we have \math{
\sigma\in\kern0.15mm^{\roman A\,\emath n}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} and such that
for all \math{\eta\in
\roman A\,n} and \math{\xi\ar 1=
\sigma\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } and \math{\xi=
\bmii8 S\fvalss20 n\fvalss01 i\fvalss10\eta } we have
\math{\xi\ar 1=\xi} if \math{\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss11\xi <
2\KPt9^{\emath n} } holds, otherwise having \mathss32{
\xi\ar 1=\Bnull_\vPi}. Then for
every \math{n\in\mathbb Z} we have \math{\bmii8 S\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} n} a
simple sequence in
\math{(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\roman A\,n\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) } with
\math{\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm(\kern0.15mm\bmii8 S\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} n\kern0.37mm)
\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} for
all \mathss30{\eta\in\roman A\,n\setminus N}, \,and in addition \math{
\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00(\kern0.15mm
\bmii8 S\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} n\fvalss01 i\kern0.37mm)
\le 2\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm } holds for all \math{n\in\mathbb Z} and \mathss36{i\in\mathbb No}. | 2,880 | 362,273 | en |
train | 0.47.54 | Letting \math{\roman E\,\sigma=
(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus\kern-0.3mm{{}^{}{\rm dom}\,{}_{{}^{}}}\sigma\kern0.37mm)\times\kern-0.2mm\{\,\Bnull_\vPi\}\cupss22
\sigma } we now take
\newline
\math{\bosy\sigma=
\seqss40{\roman E\KPt8\bigcup\KPt8\{\,
\bmii8 S\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} n\fvalss01 i:
i:n\in\mathbb Z\kern0.37mm\text{ and }\kern0.37mm|\,n\,|\suba\le i\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm\}:
i\in\mathbb No\kern-0.2mm} } thus obtaining a
simple sequence \math{\bosy\sigma} in
\math{(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } with
\math{\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy\sigma
\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} for all \mathss30{\eta\in
{}^{}\Cal Omega\kern0.07mm\setminus N} with
\mathss36{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\fvalss10\eta\not=\lower1.05mm\hbox{$^+$}\infty}, \,and such that also
\math{
\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00(\kern0.37mm\bosy\sigma\fvalss01 i\kern0.37mm)\le
2\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm } holds for all \mathss36{i\in\mathbb No}. Noting
\math{\int_{\KPp1.1{}^{}\Cal Omega\,}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu <
\lower1.05mm\hbox{$^+$}\infty } and that
from \math{\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy\sigma
\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} we get
\newline
\math{\lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty\,}(\kern0.37mm
\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10(
\kern0.37mm\bosy\sigma\fvalss01 i\fvalss10\eta - x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\svs\vPi\kern0.07mm
) = 0 } it now follows from the dominated convergence theorem
that \math{
\lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty}\kern0.15mm
\int_{\KPp1.1{}^{}\Cal Omega\,}\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10(
\kern0.37mm\bosy\sigma\fvalss01 i\fvalss10\eta - x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\svs\vPi
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm) = 0 } holds, giving the conclusion.
\end{proof} | 1,411 | 362,273 | en |
train | 0.47.55 | \begin{proposition}\label{Pro LpLp* dual}
Let \ú$\,1\le p\le\lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure on $\,
{}^{}\Cal Omega${\,\rm, }and with $\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let $\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ and $\,
F=\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \KPt8 $. Also let \inskipline0{23.4}
$\,F\aar 1=\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ or $\,
F\aar 1=\mvsLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) \, $. For \vskip.5mm\centerline{$
\beta=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\kern-0.3mm\times\kern0.15mm\ssbb00 C\capss41\{\,
(\kern0.15mm\smb X\kern0.15mm,\kern0.15mm\smb Y\kern0.07mmp,\kern0.07mm t\kern0.37mm):\aall{x\in\smb X\kern0.15mm,\kern0.15mm y\in\smb Y}\,
t=\int_{\KP1{}^{}\Cal Omega}\,y\,.\KPt8 x\rmdss11\mu\KPt9\} \KP1 $,} \inskipline{.5}0
then $\,\beta$ is a continuous bilinear map $\,F\kern0.37mm\sqcap\kern0.15mm F\aar 1\to
\bosy K$ \inskipline0{17}
with $\,\seqss40{\beta\,(\,\cdot\,,\smb Y\,) : \smb Y\kern0.15mm\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\kern-0.3mm}$
an injection.
\end{proposition}
\begin{proof} First we note that \math{\beta} is trivially a function since
the vectors of \math{F} and $\kern0.37mm F\aar 1$ \linebreak
are nonempty sets. Further, if we
know ($\kern0.15mm*\kern0.15mm$) that \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\beta } holds,
then bilinearity is readily seen. So we only need to prove ($\kern0.15mm*\kern0.15mm$)
together with continuity and the last nondegeneracy assertion. For short let \mathss36{
\roman I\KP1 x\KPt8 y = \int_{\KP1{}^{}\Cal Omega}\,y\,.\KPt8 x\rmdss11\mu}.
For ($\kern0.15mm*\kern0.15mm$) arbitrarily given \math{x\kern0.37mm,\kern0.15mm x\ar 1\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}
and \mathss31{y\kern0.37mm,\kern0.15mm y\ar 1\in\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1}, \,we need to verify
that \math{\roman I\KP1 x\KPt8 y=\roman I\KP1 x\ar 1\,y\ar 1 \in \mathbb C}
holds. For this we first note that \math{\roman I\KP1 x\KPt8 y \in\mathbb C }
under the additional assumption that \math{(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) }
is a simple mv\kern0.37mm-\kern0.15mm map. Indeed, in this case \mathss03{
\roman I\KP1 x\KPt8 y} a finite sum of expressions of the type \math{
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu } where \math{\xi
\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} and \mathss04{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+}. Noting that we
here have \vskip.5mm\centerline{$
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\KP1|\KP1 A \in
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mvLrs24^{p\sast}\kern-0.63mm(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm
\bosy K\kern0.37mm) \subseteq
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mvLrs42^1(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm
\bosy K\kern0.37mm) \KP1 $,} \inskipline{.5}0
the assertion follows. Directly from the definition we then see that \mathss30{
\roman I\KP1 x\KPt8 y=\roman I\KP1 x\KPt8 y\ar 1} holds. Then considering the
general \math{x} first with \math{p\not=\lower1.05mm\hbox{$^+$}\infty} and taking a compatible
norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \math{\vPi} and letting \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1} be the
corresponding dual norm, similarly as in the proof of Proposition \ref{Pro simp Lp dense}
above we find
a
simple sequence \math{\bosy\sigma} in
\math{(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and
some \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }
and a positive \mathss37{\mu}--\,measurable
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm} with \math{\int_{\KPp1.1{}^{}\Cal Omega\,}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu <
\lower1.05mm\hbox{$^+$}\infty } and such that
\math{
\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00(\kern0.37mm\bosy\sigma\fvalss01 i\kern0.37mm)\le
2\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm } holds for all \mathss36{i\in\mathbb No} and also
\newline
\math{\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy\sigma
\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} for all \mathss30{\eta\in
{}^{}\Cal Omega\kern0.07mm\setminus N}. Then with
\newline
\math{A=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\invss46[\KPp1.1\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.07mm\setminus\{\kern0.37mm 0\kern0.37mm\}\KP1]
\kern0.37mm\setminus N } we take
a positive \mathss37{\mu}--\,measurable \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1} with
\newline
\math{\|\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1\kern0.37mm\|\Lnorss33^{\kern0.15mm p\sast\kern-0.3mm}_\mu < \lower1.05mm\hbox{$^+$}\infty }
and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm z\KP1|\KP1 A \le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1} for
\mathss38{z\in\{\KPt8 y\kern0.37mm,\kern0.15mm y\ar 1\kern0.15mm\} }.
For
\newline
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 2=2\KP1\RHB{.3}{^p}\LHB{.2}{\kern0.15mm^{^{-1}}\kern0.37mm}
\Abrs33^p\LHB{.2}{\kern0.15mm^{^{-1}}}\KN{.5}\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\cdot\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1} now
\math{
\seqss40{z\,.\KPt8(\kern0.37mm\bosy\sigma\fvalss01 i\kern0.37mm)
\KP1|\KP1 A
:i\in\mathbb No} } converges
pointwise to
\newline
\math{z\,.\KPt8 x\KP1|\KP1 A } and is dominated by
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 2}
for which H\"older's inequality gives
\newline
\mathss36{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 2\,\|\Lnorss33^1_\mu
\le
(\,2\kern0.15mm
\int_{\KPp1.1{}^{}\Cal Omega\,}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu
\kern0.37mm)\KP1\RHB{.3}{^p}\LHB{.2}{\kern0.15mm^{^{-1}}\kern0.37mm}
\|\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1\kern0.37mm\|\Lnorss33^{\kern0.15mm p\sast\kern-0.3mm}_\mu < \lower1.05mm\hbox{$^+$}\infty
}. Consequently, by the dominated convergence theorem we obtain \inskipline{.5}{14.8}
$\roman I\KP1 x\KPt8 y=
\int_{\,A}\kern0.37mm y\,.\KPt8 x\rmdss11\mu=
\lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty\kern0.15mm}
\int_{\,A}\kern0.37mm y\,.\KPt8(\kern0.37mm\bosy\sigma\fvalss01 i\kern0.37mm)\rmdss11\mu$ \inskipline{.5}{21.75} | 3,975 | 362,273 | en |
train | 0.47.56 | $\roman I\KP1 x\KPt8 y=
\int_{\,A}\kern0.37mm y\,.\KPt8 x\rmdss11\mu=
\lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty\kern0.15mm}
\int_{\,A}\kern0.37mm y\,.\KPt8(\kern0.37mm\bosy\sigma\fvalss01 i\kern0.37mm)\rmdss11\mu$ \inskipline{.5}{21.75}
${}=
\lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty\kern0.15mm}
\int_{\,A}\kern0.37mm y\ar 1\kern0.37mm.\KPt8(\kern0.37mm\bosy\sigma\fvalss01 i\kern0.37mm)\rmdss11\mu
=
\int_{\,A}\kern0.37mm y\ar 1\kern0.37mm.\KPt8 x\rmdss11\mu
=
\roman I\KP1 x\KPt8 y\ar 1\in\mathbb C \KPt9 $. \vskip.5mm
In the case \math{p=\lower1.05mm\hbox{$^+$}\infty} we modify the above deduction as follow.
Indeed, now we have \math{p^{\,*}=1} and taking
a positive \mathss37{\mu}--\,measurable \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1} with
\math{\|\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1\kern0.37mm\|\Lnorss33^1_\mu < \lower1.05mm\hbox{$^+$}\infty }
and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm z \le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1} for
\mathss38{z\in\{\KPt8 y\kern0.37mm,\kern0.15mm y\ar 1\kern0.15mm\} } we let \mathss38{\scrmt A=
\{\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}
\kern0.07mm[\KP1 2\KP1^{\emath n\kern0.37mm - \kern0.37mm 1}\kern0.07mm,\kern0.07mm
2\KPt9^{\emath n}\kern0.37mm{[\kern0.15mm}:n\in\ssbb03 Z\,\} }. Then we find
\math{\smb M\in\lbb R_+} and \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }
such that for \math{B=\bigcup\,\scrmt A\kern0.07mm\setminus N} we have
\newline
\mathss38{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1|\KP1 B\le B\times\kern-0.2mm\{\kern0.37mm\smb M\,\} }. Now
with the notation \math{\roman E\KP1 x\,A=
(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\KPt8\Bnull_\vPi\}\cupss22
(\kern0.37mm x\KP1|\KP1 A\kern0.37mm) } a slight modification of
the above deduction gives us
\math{\roman I\KP1\roman E\KP1 x\,A\KP1 y=
\roman I\KP1\roman E\KP1 x\,A\KP1 y\ar 1 \in\mathbb C }
for all \mathss36{A\in\scrmt A}. Then again
by dominated convergence we obtain \inskipline{.4}{22}
$\roman I\KP1 x\KPt8 y = \roman I\KP1\roman E\KP1 x\,B\KP1 y =
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\roman I\KP1\roman E\KP1 x\,A\KP1 y$ \inskipline{.2}{29}
${}= \sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\roman I\KP1\roman E\KP1 x\,A\KP1 y\ar 1
= \roman I\KP1\roman E\KP1 x\,B\KP1 y\ar 1
= \roman I\KP1 x\KPt8 y\ar 1 \in \mathbb C \KPt9 $. \inskipline{.5}0
Now for the general case by the above we get \math{ \roman I\KP1 x\KPt8 y =
\roman I\KP1 x\KP1 y\ar 1 = \roman I\KP1 x\ar 1\,y\ar 1} by noting that for
some \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } we have \math{
y\,.\KPt8 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=y\,.\KPt8 x\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } for all \mathss30{
\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N}. \vskip.3mm
For continuity putting \math{ \trN03\smb X\trNu2 = \inf\kern0.15mm \big\{ \KPt8
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu\kern-0.2mm:x\in\smb X\,\} } and \inskipline{.25}{40.1}
$\trN03\smb Y\trNun5 = \inf\kern0.15mm\big\{\KPt8 \|\KP1 \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y \KP1
\|\Lnorss40^{p\sast}_\mu\kern-0.2mm : y\in\smb Y\KP1\} \KP1 $, \inskipline{.25}0
by Theorem \ref{L^p in TVS} on page \pageref{L^p in TVS} above it suffices
that we have \vskip.3mm\centerline{$
|\KPp1.2\beta\fvalss10(\kern0.15mm\smb X\kern0.07mm,\smb Y\,)\KP1|
\le\trN03\smb X\trNu2\,\trN03\smb Y\trNun5 $} \inskipline{.3}0
for \math{\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} and \math{\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1}. By Proposition \ref{Pro Hˆlder}
on page \pageref{Pro Hˆlder} for \math{x\in\smb X} and \math{y\in\smb Y} we
have \math{ |\KPp1.2\beta\fvalss10(\kern0.15mm\smb X\kern0.07mm,\smb Y\,)\KP1| \le
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu\,
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu} trivially giving the
result.
Finally, letting \math{\Bnull_{\aars F_1}\not=\smb Y\kern0.15mm\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} we
need to show existence of some \math{\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} with \linebreak \mathss03{
\beta\fvalss10(\kern0.15mm\smb X\kern0.07mm,\smb Y\,)\not=0}. Now, by \math{ \smb Y \not =
\Bnull_{\aars F_1}} there are \math{y\in\smb Y} and \math{ A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} and \mathss30{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} with \mathss36{
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu\not=0}. Then taking \math{
x=(\kern0.37mm{}^{}\Cal Omega\kern0.15mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\,\Bnull_\vPi\}\cupss22
(\kern0.15mm A\times\kern-0.2mm\{\kern0.37mm\xi\kern0.37mm\}\kern0.15mm\sbig)0 } there is \linebreak \mathss03{
\smb X} with \mathss30{x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}, \,and we now have \mathss36{
\beta\fvalss10(\kern0.15mm\smb X\kern0.07mm,\smb Y\,) =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu \not = 0 }.
\end{proof} | 3,265 | 362,273 | en |
train | 0.47.57 | \begin{corollary}\label{Coro Io inj etc}
Let \ú$\,1\le p\le\lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive
measure on $\,{}^{}\Cal Omega$ and with
\newline
$\,\bosy K
\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let $\vPi\in\roman{BaS}\kern0.4mmps0(K) \KP1 $. Also let
$\,F=\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) $ and let
\newline
$\,F\aar 1\in\{\,
\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)\,,
\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)\,,
\mvsLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)\KPt8\}$ and
$
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\kern-0.3mm\times\hbox{\font\SweD =cmssbx10\SweD U}{}\capss31\{\,(\kern0.37mm\smb Y\kern0.07mmp,\kern0.07mm\smb U
\kern0.15mm):\aall{y\in\smb Y}\,$ \inskipline{0}{37.7}
$\smb U=
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.2mm\times\mathbb C\capss31\{\,(\kern0.37mm\smb X\kern0.15mm,\kern0.07mm t\kern0.37mm) :
\aall{x\in\smb X}\,
t = \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KPt9\}\kern0.15mm\}
\KP1 $.
\noindent
Then
$\,\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\in\Cal L\,(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\dlbetss10\kern0.15mm)$ holds with
$\,\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm$ an injection.
\end{corollary}
\begin{proof} Note that although written differently, the \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} above
is precisely the same as in Theorem \nfss A\,\ref{main Th} above. Now we first
see that the assertion directly follows from Proposition \ref{Pro LpLp* dual}
above in the cases where \math{ F\aar 1 =
\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } or \mathss30{
F\aar 1 = \mvsLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } holds.
For the case \math{ F\aar 1 =
\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } putting \math{
F\aar 0=\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } and
letting \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 0} be the corresponding \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}
in the corollary, taking
\math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1=
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\kern-0.3mm\times F\aar 0\capss01\{\,(\kern0.15mm\smb X\kern0.07mm,\kern0.07mm\smb Z\kern0.15mm):
\smb X\subseteq\smb Z\,\} } we then have
\mathss34{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm=\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 0\circ\kern0.07mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1 }. Trivially having
\math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\in\Cal L\,(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\aar 0\kern0.07mm) } we get
\math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\in\Cal L\,(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\dlbetss10\kern0.15mm) } and we only
need to show that \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1} is injective. Indeed, supposing that we
have \math{x\in\smb X} and \mathss34{
(\kern0.15mm\smb X\kern0.07mm,\kern0.07mm\Bnull_{\aars F_0}\sbig)0\in
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1 }, \,for arbitrarily given \math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}
and \math{w\in\Cal L\,(\kern0.15mm\vPi\dlbetss01\KPt2,\kern0.07mm\bosy K\kern0.37mm) } we then must
show that \math{
\int_{\,A}\kern0.37mm w\circss00 x\rmdss11\mu = 0 } holds. In order to get this, we
first note that there are some \math{
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and a
separable linear subspace
\math{S\ar 1} in \math{\vPi\dlbetss01} with \mathss34{
x\KP1[\KP1 A\setminus N\KP1]\subseteq S\ar 1 }. Then from
Lemma \ref{Le 8.17.8 B} on page \pageref{Le 8.17.8 B} above we get
existence of some \math{\bosy\xi\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } with
\math{{}^{}{\rm rng}\,{}_{{}^{}}\bosy\xi\in\bouSet\vPi } and such that \math{
w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u = \lim\,(\kern0.37mm u\circss11\bosy\xi\kern0.37mm) } holds for every
\mathss34{u\in S\ar 1}. Now for all
\math{\eta\in A\setminus N} we have \vskip.25mm\centerline{$
w\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=\lim\,(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\circss11\bosy\xi\kern0.37mm)
=
\lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty\,}(\KPt5
\roman{ev}\KPt2\sbi{\bosy\xi\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i}\circ\kern0.15mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) \KP1 $.} \vskip.25mm
Since \math{x\in\smb X\subseteq\Bnull_{\aars F_0} } holds, for any fixed \math{ \xi
\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} we have \mathss36{\int_{\,A\kern0.37mm\setminus\kern0.37mm N\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.07mm x\rmdss11\mu = 0 }. From \math{ x \in
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} we see that \math{ x\KP1|\KP1 A \in
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mvLrs42^1(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) }
holds, and taking into account \math{{}^{}{\rm rng}\,{}_{{}^{}}\bosy\xi\in\bouSet\vPi } we get
existence of some positive \mathss37{\mu}--\,measurable \mathss30{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm} with \math{
\|\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\,\|\Lnorss33^1_\mu < \lower1.05mm\hbox{$^+$}\infty } and such that \math{
\Abrs03^1\kern-0.2mm\circ\kern0.15mm\roman{ev}\KPt2\sbi{\bosy\xi\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i}\circ\kern0.15mm x
\KP1|\KP1 A \le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm } holds for all \mathss36{i\in\mathbb No}. Then by dominated
convergence we obtain \inskipline{.5}{11}
$ \int_{\,A}\kern0.37mm w\circss00 x\rmdss11\mu
= \int_{\,A\kern0.37mm\setminus\kern0.37mm N\,}w\circss00 x\rmdss11\mu
= \lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty\kern0.15mm}\int_{\,A\kern0.37mm\setminus\kern0.37mm N\,}
\roman{ev}\KPt2\sbi{\bosy\xi\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i}\circ\kern0.15mm x\rmdss11\mu
= 0 \KPt8 $.
\end{proof} | 3,902 | 362,273 | en |
train | 0.47.58 | In the next lemma we utilize the formal definitions \inskipline{.6}{18}
$^{{}^{}\Cal Omega\kern0.15mm,\kern0.15mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A =
(\kern0.37mm{}^{}\Cal Omega\kern0.15mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\,\Bnull_\vPi\} \cupss22
(\kern0.15mm A\times\kern-0.2mm\{\kern0.37mm\xi\kern0.37mm\}\kern0.07mm\sbig)0
$ and \KP{18} \inskipline{.4}{18}
$\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A = \uniqset\smb X : {}
^{\bigcup\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\mu\kern0.15mm,\KPt3\vPi}\kern0.37mm\xi\kern0.15mm\sbi A \in \smb X \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \KP1 $. \vskip.5mm
If \math{\mu} is a positive measure on \mathss36{{}^{}\Cal Omega}, \,for all \math{
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+} and \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} thus \mathss30{
^{{}^{}\Cal Omega\kern0.15mm,\kern0.15mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A } is the simple function \math{{}^{}\Cal Omega\to
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} that has the value \math{\xi} at points \math{\eta\in A} and \math{
\Bnull_\vPi} else- where. Then \math{
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A } is the unique vector of \math{
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } having \math{
^{{}^{}\Cal Omega\kern0.15mm,\kern0.15mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A } as one of its representatives.
\begin{lemma}\label{Le-first}
Let $\,1\le p < \lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure on
$\,{}^{}\Cal Omega\,$. Also with $\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let
$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ with $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a compatible norm and
$\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1=\seqss44{
\sup\KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)
:u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)}$
\noindent
and $\,F=\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ and
$\,\smb U\in\Cal L\,(\kern0.15mm F\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) ${\KP1\rm, }and let
$\,(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ be finitely almost
scalarly measurable with $\,
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\ar 1\kern-0.2mm\circ\kern0.15mm y\KP1 \|\Lnorss50^{p^*}_\mu < \lower1.05mm\hbox{$^+$}\infty$
and such that
$\,
\smb U\fvalss11\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A
=\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu
$ holds for all $\,A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ and
$\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm$.
Then $\,y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm
\mvsLrs23^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ holds with
$\,\smb U=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.2mm\times\mathbb C\capss31\{\,(\kern0.37mm\smb X\kern0.15mm,\kern0.07mm t\kern0.37mm) :
\aall{x\in\smb X}\,
t = \int_{\KPp1.1{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KPt9\} \KP1 $.
\end{lemma}
\begin{proof} We get \math{y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm
\mvsLrs23^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } directly from the
definition, and hence
only the last formula has to be verified. To get this, we note that for \math{
\smb X} and \math{x} with \math{{}^{}{\rm rng}\,{}_{{}^{}} x} finite and \mathss30{
x \in \smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}, \,i.e.\ for some finite function \math{ \scrmt S
\subseteq (\kern0.37mm\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.07mm\sbig)0\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } with \math{
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt S} disjoint and \math{x=
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.15mm]_{vs}\,\text{-}\sum_{\,A\kern0.37mm\in\kern0.37mm
{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S\,}
{}^{{}^{}\Cal Omega\kern0.15mm,\kern0.15mm\vPi}\kern0.37mm
(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\kern0.15mm\sbi A } we trivially have \inskipline{.5}{20}
$ \smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X
= \sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S}\kern0.15mm
\int_{\,A}\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)
= \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu $ \inskipline{.7}0
and by Proposition \ref{Pro Hˆlder} with \math{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm = \seqss33{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\cdot(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss11 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) :
\eta = \eta} } we get \inskipline1{19.1}
$ \big|\kern0.15mm\int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KP1|
= \big|\,\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S}\kern0.15mm
\int_{\,A}\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)\KP1|$ \inskipline{.7}{41}
${}\le\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S}\kern0.15mm\int_{\,A\,}|\KP{1.2}
y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm) \KP{1.1} |
\rmdss21\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline{.7}{41} | 3,818 | 362,273 | en |
train | 0.47.59 | $ \smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X
= \sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S}\kern0.15mm
\int_{\,A}\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)
= \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu $ \inskipline{.7}0
and by Proposition \ref{Pro Hˆlder} with \math{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm = \seqss33{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\cdot(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss11 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) :
\eta = \eta} } we get \inskipline1{19.1}
$ \big|\kern0.15mm\int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KP1|
= \big|\,\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S}\kern0.15mm
\int_{\,A}\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)\KP1|$ \inskipline{.7}{41}
${}\le\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S}\kern0.15mm\int_{\,A\,}|\KP{1.2}
y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm) \KP{1.1} |
\rmdss21\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline{.7}{41}
${}\le\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S}\kern0.15mm\upint\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1(\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1|\KP1 A\kern0.37mm)\rmdss11\mu$ \inskipline{.7}{41}
${}\le \upint\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu
\le \|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu\,
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss11 x\KP1\|\Lnorss33^p_\mu \KP1 $. \inskipline10
Since by Proposition \ref{Pro simp Lp dense} on page \pageref{Pro simp Lp dense}
above the set of vectors with simple representatives is \mathss35{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F
}--\,dense, from Corollary \ref{Coro Io inj etc} it follows that \math{
\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X = \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu }
holds for all \math{x\kern0.37mm,\kern0.15mm\smb X} with \math{x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}, \,and
this is precisely what we needed.
\end{proof} | 1,950 | 362,273 | en |
train | 0.47.60 | \begin{proposition}\label{Pro L^1'=L^i}
Let $\,\mu$ be an almost decomposable positive measure on $\,{}^{}\Cal Omega${\,\rm, }
and with $\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let $\,F=\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$
and $\,F\aar 1=\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ and \vskip.5mm\centerline{$
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1=\seq{ \KP{1.2} \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.2mm\times\mathbb C\capss31\{\,
(\kern0.37mm\smb X\kern0.15mm,\kern0.07mm t\kern0.37mm) : \aall{x\in\smb X\kern0.15mm,\kern0.15mm y\in\smb Y}\,
t = \int_{\KP{1.1}{}^{}\Cal Omega\,}x\cdot y\rmdss11\mu\KPt9\} :
\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\, } \KP1 $.} \inskipline{.5}0
Then $\,\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\in\Lis(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\dlbetss10\kern0.15mm)$ holds.
\end{proposition}
\begin{proof} Taking \math{p=1} and \math{\vPi=\bosy K} in Corollary \ref{Coro Io inj etc}
above, we see that $\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1$ is a continuous linear injection \mathss35{
F\aar 1\to F^{\kern0.4mm\prime}_{\kern-.2mm\raise.95mm\hbox{$_{_\beta}$}}}. Since \math{F} is normable by Theorem \ref{L^p in TVS}
above, by Corollary \ref{Cor L^p Ban} the spaces \math{F\aar 1} and \math{
F\dlbetss10} are \erm Banachable, and so by the open mapping theorem we only
need to prove that \math{ \Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) \subseteq
{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1 } holds. To establish this, arbitrarily fixing \mathss37{
\smb U\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }, \,let
$\smb M=\sup\kern0.15mm\big\{\KPt8|
\KP{1.1}\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X\KPt9|:
\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.37mm\text{ and }\kern0.37mm\aall{x\in\smb X}\,
\int_{\KP{1.1}{}^{}\Cal Omega\,}
|\KP1 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\rmdss11\mu\,(\kern0.07mm\eta\kern0.07mm)
\le 1\KPt9\}$ ,
and let $\scrmt A$ and $N\kern0.15mmrim1$ be as in
Definitions \ref{df decomp}\,(2) on page \pageref{decos A} above.
Then $\smb M\in\lbb R_+$ holds, and we let $\scrmt Y\ar 1$ be the set of all
pairs $(\kern0.15mm A\ar 1\kern0.15mm,\kern0.07mm y\ar 1)$ with $A\ar 1\in\scrmt A$ and
$y\ar 1\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm
\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu
\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) \,$ and $\,
\sup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}(\kern0.37mm\Abrs00^1\circ\kern0.15mm y\ar 1) \le \smb M \,$ and
such that
$\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X=
\int_{\,\aars A_1}x\cdot y\ar 1\rmdss01\mu$
holds for all $x\kern0.37mm,\kern0.15mm\smb X$ with $x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F$ and
$x\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\,\ssbb52 C\setminus\{\kern0.37mm 0\kern0.37mm\}\KP{1.1}]\subseteq A\ar 1
\kern0.37mm$.
Then from \cite[Theorem 6.4.1\kern0.15mm, p.\ 162]{Du} we know that
$\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y\ar 1$ holds, and hence by the {\sl axiom of choice\kern0.15mm}
there is a function $\scrmt Y\subseteq\scrmt Y\ar 1$ with
$\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y\ar 1\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y\,$. Taking \math{y=
N\kern0.15mmrim1\kern-0.3mm\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss24\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\scrmt Y}, \,by
Lemma \ref{Le deco meas} on page \pageref{Le deco meas}
above \math{(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } now
\linebreak
is finitely almost measurable, and hence
\math{y\in\smb Y} holds for some \mathss31{\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1}.
Then for given \math{x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} letting
\mathss38{\scrmt A\kern0.15mm\ar 0=\scrmt A\capss31\{\,A:
\int_{\,A\kern0.37mm}\Abrs00^1\circ\kern0.15mm x\rmdss11\mu\not=0\KP1\} }, \,we have
\math{\scrmt A\kern0.15mm\ar 0} countable. If \math{\scrmt A\kern0.15mm\ar 0} is infinite,
we take any bijection \mathss36{\ebit A:\mathbb No\to\scrmt A\kern0.15mm\ar 0}, \,and if
it is finite, for some
\math{\smb N\in\mathbb No} we first take a
bijection \mathss36{\ebit A\ar 0\kern-0.2mm:\smb N\to\scrmt A\kern0.15mm\ar 0} and then
put \math{\ebit A=(\kern0.37mm\mathbb No\kern-0.3mm\setminus\smb N\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm
\emptyset\kern0.37mm\}\cupss21\ebit A\ar 0}.
Let now
\math{\ebit B=\kern0.15mm\big\langle\,\bigcup\KP1(
\kern0.15mm\ebit A\KPt8|\KP1 i\kern0.37mm):i\in\mathbb No\,\rangle} and
$\roman x\,i=(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus(\kern0.37mm\ebit B\fvalss01 i\kern0.37mm))\times\kern-0.2mm
\{\kern0.37mm 0\kern0.37mm\}\cupss22
(\kern0.37mm\ebit B\fvalss01 i
\times\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0 \,$ and
$\bosy x=
\seqss30{(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus(\kern0.15mm\ebit A\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\kern0.37mm))\times\kern-0.2mm
\{\kern0.37mm 0\kern0.37mm\}\cupss22(\kern0.37mm x\KP1|\KP1(\kern0.15mm\ebit A\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\kern0.37mm)):
i\in\mathbb No}$
and \math{\ebit X=
\seqss30{\uniqset\smb X\kern-0.3mm:\bosy x\fvalss01 i\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.3mm:
i\in\mathbb No} }
and
\mathss39{\ebit Y\kern0.37mm=
\seqss33{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\KPt8\text{-\kern0.15mm}\sum\KP1(\kern0.15mm\ebit X\KPt8|\KP1 i\kern0.37mm)
:i\in\mathbb No} }.
Then we have
\math{\ebit Y\kern0.37mm\to\smb X} in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F} and hence also
\math{\smb U\circ\ebit Y\to
\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X}. Consequently, by
dominated convergence we obtain \inskipline1{11}
$\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}\,(\kern0.37mm\smb U\circ\ebit Y\fvalss81 i\kern0.37mm)
=
\lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}\sum_{\KPt8 k\kern0.37mm\in\kern0.37mm i\,}
(\kern0.37mm\smb U\circ\ebit X\fvalss21 k\kern0.37mm) $ \inskipline1{18}
${}
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}\sum_{\KPt8 k\kern0.37mm\in\kern0.37mm i\kern0.15mm}
\int_{\,\bmii6 A\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} k\,}x\cdot y\rmdss11\mu
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}
\int_{\KPt8\bmii6 B\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\,}x\cdot y\rmdss11\mu $ \inskipline1{18} | 4,030 | 362,273 | en |
train | 0.47.61 | $\roman x\,i=(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus(\kern0.37mm\ebit B\fvalss01 i\kern0.37mm))\times\kern-0.2mm
\{\kern0.37mm 0\kern0.37mm\}\cupss22
(\kern0.37mm\ebit B\fvalss01 i
\times\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0 \,$ and
$\bosy x=
\seqss30{(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus(\kern0.15mm\ebit A\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\kern0.37mm))\times\kern-0.2mm
\{\kern0.37mm 0\kern0.37mm\}\cupss22(\kern0.37mm x\KP1|\KP1(\kern0.15mm\ebit A\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\kern0.37mm)):
i\in\mathbb No}$
and \math{\ebit X=
\seqss30{\uniqset\smb X\kern-0.3mm:\bosy x\fvalss01 i\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.3mm:
i\in\mathbb No} }
and
\mathss39{\ebit Y\kern0.37mm=
\seqss33{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\KPt8\text{-\kern0.15mm}\sum\KP1(\kern0.15mm\ebit X\KPt8|\KP1 i\kern0.37mm)
:i\in\mathbb No} }.
Then we have
\math{\ebit Y\kern0.37mm\to\smb X} in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F} and hence also
\math{\smb U\circ\ebit Y\to
\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X}. Consequently, by
dominated convergence we obtain \inskipline1{11}
$\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}\,(\kern0.37mm\smb U\circ\ebit Y\fvalss81 i\kern0.37mm)
=
\lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}\sum_{\KPt8 k\kern0.37mm\in\kern0.37mm i\,}
(\kern0.37mm\smb U\circ\ebit X\fvalss21 k\kern0.37mm) $ \inskipline1{18}
${}
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}\sum_{\KPt8 k\kern0.37mm\in\kern0.37mm i\kern0.15mm}
\int_{\,\bmii6 A\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} k\,}x\cdot y\rmdss11\mu
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}
\int_{\KPt8\bmii6 B\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\,}x\cdot y\rmdss11\mu $ \inskipline1{18}
${}
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}
\int_{\KPp1.1{}^{}\Cal Omega\,}x\cdot y\cdot\roman x\,i\rmdss11\mu
= \int_{\KPp1.1{}^{}\Cal Omega\,}x\cdot y\rmdss11\mu
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.37mm\smb Y\KPt8\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb X\kern0.37mm$.
\end{proof} | 1,519 | 362,273 | en |
train | 0.47.62 | For a topology \math{\scrmt T} we say that that \math{\scrmt T} is
{\it separably metrizable\kern0.37mm} if{}f \math{\scrmt T} is a metrizable topology
and there is a countable \math{D\subseteq\bigcup\,\scrmt T} with \mathss34{
\bigcup\,\scrmt T\subseteq\roman{Cl\KPt8}\sbi{\scrm7 T\KPt8}D }. In particular then \math{
D} is \mathss37{\scrmt T}--\,dense. Now, for the purpose of Lemma \ref{Le |int| < M imp ...}
below we put the following
\begin{definitions}\label{df sep cnv metr}
(1) \ Say that \math{C} is {\it separably uniform metrizable\kern0.37mm} in \math{E}
if{}f \math{E} is a real or complex topological vector space and there are
some nonempty countable sets \mathss03{D\kern0.37mm,\kern0.15mm\scrmt U} with \math{D\subseteq C
\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} and \math{\scrmt U\subseteq\neiBoo E} and such that \math{D} is \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C}--\,dense and for every \math{x\in C} it holds that \math{
\big\{\,[\KP1\{\kern0.37mm x\kern0.37mm\} +\kern0.15mm U\KPp1.1]\svs E\capss13 C : \kern0.15mm U \kern-0.3mm \in
\scrmt U\KP1\} } is a filter base for \mathss38{
\Cal N_{\font\SweD =cmmi6\lower.15mm\hbox{\kern.1mm\SweD bh\kern.15mm}}(\kern0.37mm x\kern0.37mm,\kern0.07mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C\kern0.37mm) }, \inskipline{.5}2
(2) \ Say that \math{E} is {\it countably separably convex metrizable\kern0.37mm}
if{}f \math{E} is a real or complex Hausdorff locally convex space and
there is a countable \math{\scrmt C} with \math{ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E = \bigcup\,\scrmt C}
and such that \math{C} is separably uniform metrizable in \mathss03{E} for
every \mathss34{C\in\scrmt C}.
\end{definitions}
Examples of countably separably convex metrizable spaces are all locally
convex spaces \math{E} with \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E} separably metrizable as well as
countable strict inductive limits of such spaces. In particular, for example \math{
\mathscr D\,(\ssbb43 R) } and \math{C\kern.6mm\raise.15mm\hbox{$^\infty$}\kern.07mm(\ssbb43 R) } are countably
separably convex metrizable. Also \math{\vPi\dlsigss00\kern0.07mm} is countably
separably convex metrizable when \math{\vPi} is normable with \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} a separable topology.
Note that by the metrization theorem \cite[6.13\kern0.37mm, p.\ 186]{Ky} the
\q{uniform} filter base condition in Definitions \ref{df sep cnv metr}\,(1)
implies that \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C} is a metrizable topology. We leave it
as an {\sl open problem\kern0.15mm} whether we would have obtained an equivalent
definition if in \ref{df sep cnv metr}\,(1) instead of that uniformity
condition we had just required \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C} to be a metrizable
topology. We also remark that the definition given above is precisely what we
need in the next
\begin{lemma}\label{Le |int| < M imp ...}
Let $\,\vPi$ be countably separably convex metrizable{\kern0.15mm\rm, }and let $\,C$
be closed and convex in $\,\vPi\kern0.15mm$. Also let $\,
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ be finitely scalarly integrable and such
that \vskip.4mm\centerline{$
\int_{\,A\,}u\circss00 x\rmdss11\mu\in\{\KPt8\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\cdot t :
t\in u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm C\KP1\} $} \inskipline{.4}0
for $\, A \in \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ and $\, u \in
\Cal L\,(\kern0.15mm\vPi\Reit2\kern0.15mm,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) \KPt8 $. Then $\,
x\invss46[\KP1\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C\KP1]$ is $\,\mu\,$--\,negligible.
\end{lemma}
\begin{proof} Let \math{\scrmt C} be as in
Definitions \ref{df sep cnv metr}\,(2) above when in place of \math{E} we have
taken the \math{\vPi} in the lemma. Then taking into account
(\kern0.15mm\erm{NB}\,2\kern0.15mm) in \cite[p.\ 33]{Jr} by {\sl dependent choice\kern0.15mm} we find
countable sets \math{D\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} and \math{\scrmt P\subseteq
\scrmt C\times(\kern0.37mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\capss12\neiBoo\vPi\kern0.37mm)} with
\math{\scrmt C\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt P } and such that for \math{
(\kern0.37mm C\ar 1\kern0.15mm,\kern0.15mm U\kern0.37mm)\in\scrmt P } and
\math{\scrmt U=\scrmt P\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\,C\aar 1\} } it holds that
\math{U} is absolutely \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}--\,convex and there is
\math{V\in\scrmt U} with
\math{[\KPp1.1 V\kern-0.3mm + \kern0.15mm V\KPp1.1]\svs\vPi\subseteq U}
and
also \math{C\ar 1\kern-0.2mm\cap\KPt3 D} is \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C\ar 1}--\,dense and
for every \math{\xi\in C\ar 1} it holds that
\newline
\math{
\big\{\,[\KP1\{\kern0.37mm\xi\kern0.37mm\} +\kern0.15mm V\KPp1.1]\svs\vPi\capss13 C\ar 1\kern-0.3mm :
\kern0.15mm V \kern-0.3mm \in
\scrmt U\KP1\} } is a filter base for \mathss38{
\Cal N_{\font\SweD =cmmi6\lower.15mm\hbox{\kern.1mm\SweD bh\kern.15mm}}(\,\xi\,,\kern0.07mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C\ar 1\kern0.07mm) }.
Now we let \math{\scrmt R} be the countable set of all triplets \math{
(\kern0.37mm C\ar 1\kern0.15mm,\kern0.07mm\xi\,,\kern0.07mm U\aar 1) } such that there is
\math{U} with \math{
(\kern0.37mm C\ar 1\kern0.15mm,\kern0.15mm U\kern0.37mm)\in\scrmt P } and
\math{\xi\in C\ar 1\kern-0.2mm\cap\KPt3 D } and
\math{U\aar 1=[\KPp1.1\{\kern0.37mm\xi\kern0.37mm\} + \kern0.15mm U\KPp1.1]\svs\vPi } and
\mathss36{C\capss23 U\aar 1=\emptyset}. Then by
{\sl Hahn\,--\,Banach\kern0.15mm} \cite[7.3.2\kern0.37mm, p.\ 130]{Jr} in conjunction
with {\sl countable choice\kern0.15mm} we get existence of a function
\math{\scrmt R\to\Cal L\,(\kern0.15mm\vPi\Reit2\kern0.15mm,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) } with the
property that
\newline
\math{\sup\,(\kern0.37mm u\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} C\kern0.37mm) < u\fvalss01\xi\ar 1}
holds for
\math{(\kern0.37mm C\ar 1\kern0.15mm,\kern0.07mm\xi\,,\kern0.07mm U\aar 1\kern0.15mm,\kern0.07mm u\kern0.37mm)\in
\scrmt S} and \mathss32{\xi\ar 1\in\kern0.15mm U\aar 1}. | 3,576 | 362,273 | en |
train | 0.47.63 | Now we let \math{\scrmt R} be the countable set of all triplets \math{
(\kern0.37mm C\ar 1\kern0.15mm,\kern0.07mm\xi\,,\kern0.07mm U\aar 1) } such that there is
\math{U} with \math{
(\kern0.37mm C\ar 1\kern0.15mm,\kern0.15mm U\kern0.37mm)\in\scrmt P } and
\math{\xi\in C\ar 1\kern-0.2mm\cap\KPt3 D } and
\math{U\aar 1=[\KPp1.1\{\kern0.37mm\xi\kern0.37mm\} + \kern0.15mm U\KPp1.1]\svs\vPi } and
\mathss36{C\capss23 U\aar 1=\emptyset}. Then by
{\sl Hahn\,--\,Banach\kern0.15mm} \cite[7.3.2\kern0.37mm, p.\ 130]{Jr} in conjunction
with {\sl countable choice\kern0.15mm} we get existence of a function
\math{\scrmt R\to\Cal L\,(\kern0.15mm\vPi\Reit2\kern0.15mm,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) } with the
property that
\newline
\math{\sup\,(\kern0.37mm u\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} C\kern0.37mm) < u\fvalss01\xi\ar 1}
holds for
\math{(\kern0.37mm C\ar 1\kern0.15mm,\kern0.07mm\xi\,,\kern0.07mm U\aar 1\kern0.15mm,\kern0.07mm u\kern0.37mm)\in
\scrmt S} and \mathss32{\xi\ar 1\in\kern0.15mm U\aar 1}.
Now taking \mathss38{ \scrmt O = \{\,u\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\openIval{\sup\,(\kern0.37mm
u\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} C\kern0.37mm)\,,\lower1.05mm\hbox{$^+$}\infty\kern0.15mm}:u\in{}^{}{\rm rng}\,{}_{{}^{}}\scrmt S\KPt8\} }, \,we have \mathss30{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C={\kern-0.63mm}} \mathss04{\bigcup\,\scrmt O}. Indeed,
trivially \math{\bigcup\,\scrmt O\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C } holds, and for
the converse inclusion arbitrarily fixing \math{ \xi\ar 0 \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C} we first find some \math{C\ar 1}
with \mathss34{\xi\ar 0\in C\ar 1\in \scrmt C}. Then we find
\math{U\in\scrmt P\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\,C\aar 1\} } such that for
\math{U\aar 0=[\KPp1.1\{\,\xi\ar 0\kern0.15mm\} + \kern0.15mm U\KPp1.1]\svs\vPi }
we have \mathss36{C\capss23 U\aar 0=\emptyset}. We further find
\math{V\in\scrmt P\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\,C\aar 1\} } with
\math{[\KPp1.1 V\kern-0.3mm + \kern0.15mm V\KPp1.1]\svs\vPi\subseteq U}
and then there is some \math{\xi\in C\ar 1\kern-0.2mm\cap\KPt3 D }
with \mathss30{(\,\xi - \xi\ar 0\kern0.07mm)\svs\vPi\in\kern0.15mm V}. Now
putting \math{U\aar 1=[\KPp1.1\{\kern0.37mm\xi\kern0.37mm\} + \kern0.15mm V\KPp1.1]\svs\vPi }
we have \math{(\kern0.37mm C\ar 1\kern0.15mm,\kern0.07mm\xi\,,\kern0.07mm U\aar 1)
\in\scrmt R={{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt S } and hence there is
\math{u} with \mathss34{
(\kern0.37mm C\ar 1\kern0.15mm,\kern0.07mm\xi\,,\kern0.07mm U\aar 1\kern0.15mm,\kern0.07mm u\kern0.37mm)\in\scrmt S }. Noting
that now \math{\xi\ar 0\in U\aar 1\subseteq
u\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\openIval{\sup\,(\kern0.37mm
u\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} C\kern0.37mm)\,,\lower1.05mm\hbox{$^+$}\infty\kern0.15mm}
} holds,
we obtain \mathss34{\xi\ar 0\in\bigcup\,\scrmt O }.
Now, to prove that \math{x\invss46[\KP1\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C\KP1] } is \mathss37{
\mu}--\,negligible, arbitrarily fixing
\newline
\mathss30{A\kern0.15mm\ar 0\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}, \,by
{\sl countable choice\kern0.15mm} and the discussion after
the proof of Lemma \ref{Le +int} on page \pageref{int not meas} we find
\math{N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and
a function \math{\varPhi:{}^{}{\rm rng}\,{}_{{}^{}}\scrmt S\to
\kern0.15mm^{\aars A_0}\,\mathbb R} such that \math{
(\kern0.37mm\varphi\,;\kern0.07mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.15mm\ar 0\,,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)
} is measurable and such that
\math{u\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta} holds for
\math{(\kern0.37mm u\kern0.37mm,\kern0.07mm\varphi\kern0.37mm)\in\varPhi} and
\mathss32{\eta\in A\kern0.15mm\ar 0\kern-0.3mm\setminus N\aar 1}. For
\math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.15mm\ar 0} we then also have
\newline
\mathss38{
\int_{\,A}\kern0.37mm\varphi\rmdss21\mu=
\int_{\,A\,}u\circss00 x\rmdss11\mu\in
\{\KPt8
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\cdot t:t\in u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm C\KP1\} }. Now
\math{u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm C} is a real interval and hence for
\math{N\kern0.15mmrim1=\varphi\invss44
[\,\ssbb42 R\setminus u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm C\KPp1.1] } we have
\mathss36{N\kern0.15mmrim1\in
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.15mm\ar 0}. Since \math{0 < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm N\kern0.15mmrim1}
would
trivially give a contradiction, we in fact have
\mathss38{N\kern0.15mmrim1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
Then {\sl countable choice\kern0.15mm} gives us existence of
\math{N} with
\math{N\aar 1\subseteq N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and such that
\newline
\math{\varphi\KPp1.1[\KP1 A\kern0.15mm\ar 0\kern-0.3mm\setminus N\KP1]\subseteq
u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm C} holds for
\mathss34{(\kern0.37mm u\kern0.37mm,\kern0.07mm\varphi\kern0.37mm)\in\varPhi}. It being a trivial
exercise to check that now \math{
x\invss46[\KP1\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C\KP1]\capss41 A\kern0.15mm\ar 0
\subseteq N} holds, we are done.
\end{proof} | 3,411 | 362,273 | en |
train | 0.47.64 | From Lemma \ref{Le |int| < M imp ...} we obtain the following immediate
\begin{corollary}\label{Coro |f|<M}
Let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega$ with \ú$\,\mu\fvalss01{}^{}\Cal Omega <
\lower1.05mm\hbox{$^+$}\infty${\,\rm, }and with \ú$\,\smb M\in\lbb R_+$
and
\ú$\,\varphi\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs42^1(\kern0.37mm\mu\,,\kern-0.2mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm) $ let \ú$\,
\big|\kern0.15mm\int_{\,A}\kern0.37mm\varphi\rmdss11\mu\KP1|\le\smb M
\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)$ hold for all
\ú$\,A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$. Then there is
$\,N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that $\,
|\KP1\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\le\smb M$
holds for all $\,\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.07mm$.
\end{corollary}
\begin{lemma}\label{Le L^1_si-compa}
Let
$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ and let $\,\mu$ be a positive measure with
$\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu < \lower1.05mm\hbox{$^+$}\infty\,$. Also let
$\,K\in
\bouSet
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) \KP1$. Then $\,K$ is relatively
$\,
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\sbig(3\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\subsigma\kern0.07mm)\,
$--\,compact if and only if
for every $\,\varepsilon\in\rbb R^+$ there is $\,\delta\in\rbb R^+$
such that $\,\|\KP1\varphi\KP1|\KP1 A\KP1\|\Lnorss33^1_\mu
<\varepsilon$ holds for all
$\,\varphi\in\bigcup\,K$ and $\,
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm[\KPp1.1 0\,,\kern0.07mm\delta\KP1{[}
\KP1$.
\end{lemma}
\begin{proof} \newcommand\sFsigmaprime{F^{\kern.2mm\prime}_{\kern-.2mm\sigma}}
The assertion is already in
\cite[Theorem 3.2.1\kern0.15mm, p.\ 376]{Du-pe}\,, although one should note that
\q{weakly compact} there means \q{relatively weakly sequentially compact}. To
get a proper proof, suitably adapt the proof of
\cite[Theorem 4.21.2\kern0.37mm, pp.\ 274\,--\,275]{Edw}\,. Since we shall below need
the \q{if\kern0.15mm} part, we here give an explicit proof of it. Indeed, letting
(\kern0.15mm$*$\kern0.15mm) denote the asserted sufficient condition, and putting \mathss03{E=
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } and \mathss38{F=
\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }, \,let \mathss30{ \twEps =
\seqss43{\roman{ev}\kern0.37mm\sbi{\ssmb\Phii}\KPt8|\KP1\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm
\bosy K\kern0.37mm):\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} } \linebreak
and \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm=\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2\circ\kern0.15mm\twEps}
where \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2:E\dlbetss12\kern-0.63mm\dlbetss01\to F\dlbetss10} is the
transpose of \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.2mm:F\to E\dlbetss11} when \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1} is
as in Proposition \ref{Pro L^1'=L^i} on page \pageref{Pro L^1'=L^i} above.
Then \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is a strict morphism \math{E\subsigrs04\to F\dlsigss10} in
the sense of \cite[Definition 2.5.1\kern0.37mm, p.\ 100]{Ho}\,. Now assuming that
(\kern0.15mm$*$\kern0.15mm) holds, since by Alaoglu's theorem from \math{K\in\bouSet E} we
know that \math{\Cl_taurd{(\sFsigmaprime)}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K\kern0.37mm) } is \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm F\dlsigss00\kern0.15mm)}--\,compact, it suffices to prove that \math{
\Cl_taurd{(\sFsigmaprime)}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm } holds.
Thus arbitrarily given \math{w\in
\Cl_taurd{(\sFsigmaprime)}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K\kern0.37mm) } with \math{{}^{}\Cal Omega=
\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \vskip.3mm\centerline{$
\roman x\,A=(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22
(\kern0.15mm A\times\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0$} \inskipline{.3}0
and \math{\eightroman X\,A=\uniqset\smb\Psii:\roman x\,A\in\smb\Psii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}
putting \math{\lambda=\seqss33{w\fvalss02\text{\erm X\,}A:A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} } we
see that now \math{\lambda} is a real or complex measure that is absolutely \mathss37{
\mu}--\,continuous.
Indeed, given \math{\varepsilon\in\rbb R^+} by (\kern0.15mm$*$\kern0.15mm) there is \math{ \delta \in
\rbb R^+} such that for all \math{ A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm[\KPp1.1 0\,,\kern0.07mm\delta\KP1{[} } we have \math{
|\KP1 z\fvalss02\text{\erm X\,}A\KP1|\le\varepsilon } for all \math{ z \in
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K} and hence also \math{|\KP1\lambda\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1| =
|\KP1 w\fvalss02\text{\erm X\,}A\KP1| \le \varepsilon } holds.
Note that \math{\lambda} is trivially finitely additive, and that countable
additivity then follows from the established absolute continuity. Now by
Radon\,--\,Nikodym there is some \mathss03{\varphi\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} with \math{
w\fvalss02\text{\erm X\,}A = \lambda\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A =
\int_{\,A\kern0.15mm}\varphi\rmdss11\mu } for all \mathss36{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}. Then there
is \mathss30{\smb\Phii} with \mathss35{\varphi\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E}, \,and
noting that the linear \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}--\,span of \math{
\{\KPt8\text{\erm X\,}A:A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\KPt8\} } is \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}--\,dense,
we first see that \mathss30{w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii=
\int_{\KPp1.1{}^{}\Cal Omega\,}\varphi\cdot\psi\rmdss11\mu} holds for \mathss31{\psi\in
\smb\Psii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}. Then we get \math{w=\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\in{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}
from \inskipline{.5}{19.6} | 4,074 | 362,273 | en |
train | 0.47.65 | \roman x\,A=(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22
(\kern0.15mm A\times\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0$} \inskipline{.3}0
and \math{\eightroman X\,A=\uniqset\smb\Psii:\roman x\,A\in\smb\Psii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}
putting \math{\lambda=\seqss33{w\fvalss02\text{\erm X\,}A:A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} } we
see that now \math{\lambda} is a real or complex measure that is absolutely \mathss37{
\mu}--\,continuous.
Indeed, given \math{\varepsilon\in\rbb R^+} by (\kern0.15mm$*$\kern0.15mm) there is \math{ \delta \in
\rbb R^+} such that for all \math{ A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm[\KPp1.1 0\,,\kern0.07mm\delta\KP1{[} } we have \math{
|\KP1 z\fvalss02\text{\erm X\,}A\KP1|\le\varepsilon } for all \math{ z \in
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K} and hence also \math{|\KP1\lambda\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1| =
|\KP1 w\fvalss02\text{\erm X\,}A\KP1| \le \varepsilon } holds.
Note that \math{\lambda} is trivially finitely additive, and that countable
additivity then follows from the established absolute continuity. Now by
Radon\,--\,Nikodym there is some \mathss03{\varphi\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} with \math{
w\fvalss02\text{\erm X\,}A = \lambda\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A =
\int_{\,A\kern0.15mm}\varphi\rmdss11\mu } for all \mathss36{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}. Then there
is \mathss30{\smb\Phii} with \mathss35{\varphi\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E}, \,and
noting that the linear \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}--\,span of \math{
\{\KPt8\text{\erm X\,}A:A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\KPt8\} } is \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}--\,dense,
we first see that \mathss30{w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii=
\int_{\KPp1.1{}^{}\Cal Omega\,}\varphi\cdot\psi\rmdss11\mu} holds for \mathss31{\psi\in
\smb\Psii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}. Then we get \math{w=\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\in{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}
from \inskipline{.5}{19.6}
$ w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii
= \int_{\KPp1.1{}^{}\Cal Omega\,}\varphi\cdot\psi\rmdss11\mu
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii
= \twEps\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii\kern0.15mm) $ \inskipline{.4}{26}
${}
= \twEps\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\circss00\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\twEps\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2\circ\twEps\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii\,$.
\end{proof} | 2,392 | 362,273 | en |
train | 0.47.66 | \Ssubhead C Lifting and integral representations \label{Sec C}
As auxiliary results for the proof of Theorem \nfss A\,\ref{main Th} we
reformulate some forms of the Dunford\,--\,Pettis theorem
\cite[8.17.6\,--\,8\kern0.37mm, p.\ 584]{Edw} in Propositions \ref{Pro Edw 8.17.6}
and \ref{Pro Edw 8.17.8} below. The essential content of
\cite[Lemma 8.17.1\,(a)\,, p.\ 579]{Edw} is in the following
\begin{proposition}\label{Pro lift}
Let $\,\mu$ be an almost decomposable positive measure on $\,{}^{}\Cal Omega
${\,\rm, }and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ and \ú$\,
G=\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ let $\,S$ be a
vector subspace in $\,\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm G$ such that $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm G\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 S$ is a
separable topology. \vskip.2mm\centerline{
Then a choice function $\,c\in
\Cal L\,(\kern0.37mm G_{\kern0.15mm/\kern0.37mm S}\kern0.37mm,\kern0.15mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm
(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm))$ exists.}
\end{proposition}
\begin{proof} Letting
\math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm G_{\kern0.37mm|\kern0.37mm S}=(\kern0.37mm a\,,\kern0.15mm c\kern0.37mm)} and
\math{R=
\{\,s + t\KP1\imag:s\kern0.37mm,\kern0.15mm t\in\ssbb04 Q\,\} } we first put
\ú$\kern0.37mm X={\kern-0.63mm}$
\linebreak
\ú$(\kern0.37mm a\,,\kern0.15mm c\KP1|\KP1(\kern0.15mm R\times S\kern0.37mm)) \kern0.37mm$ and
consider vector subspaces in the
possibly complex rational vector space \mathss31{X}.
Thus, letting \math{D} be countable and \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm G\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 S
}--\,dense,
let $S\ar 1$ be the linear
\mathss37{X}--\,span of $D$. Then let $B$ be a linear basis of $
X_{\kern0.37mm|\kern0.37mm\aars S_1}$.
By {\sl countable choice\kern0.15mm} there is a choice function
$c\ar 0$ of $B$, and we let $\bar c\ar 0$ be its unique linear extension $
X_{\kern0.37mm|\kern0.37mm\aars S_1}\to
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs}\kern0.37mm_{|
\,\bigcup\,S} \,$. Letting $\scrmt A$ and $N$ be as in
Definitions \ref{df decomp}\,(2) on page \pageref{decos A} above,
for every fixed $A\in\scrmt A$ we then see existence of some $N\aar 1\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
$|\KP{1.2}\bar c\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\le
\sup\kern0.37mm\big\{\,|\KP1\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|:
\eta\in A\KP1\}$ holds for $\varphi\in\smb\Phii\in S\ar 1$ and
$\eta\in A\kern0.15mm\setminus N\aar 1\kern0.37mm$. Then by
the {\sl axiom of choice\kern0.15mm} from the property of being almost
decomposable we see existence of a \mathss37{\mu}--\,negligible
$N\kern0.07mm{}^{{}_{{}^{'\!}}}$ such that
$|\KP{1.2}\bar c\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\le
\sup\kern0.37mm\big\{\,|\KP1\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|:
\eta\in{}^{}\Cal Omega\KP1\}$ holds for
$\varphi\in\smb\Phii\in S\ar 1$ and
$\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.07mm{}^{{}_{{}^{'\!}}}\kern0.15mm$. Now taking
\mathss38{c\ar 1=
\seqss40{
\bar c\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\KP1|\KP1(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.07mm{}^{{}_{{}^{'\!}}}\kern0.07mm
)\cupss22(\kern0.15mm N\kern0.07mm{}^{{}_{{}^{'\!}}}\kern-.2mm\times\kern-.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\sbig)0
:\smb\Phii\in S\ar 1} }, \,we have $c\ar 1$ a
linear map $X_{\kern0.37mm|\kern0.37mm\aars S_1}\to
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm) $
and also
\math{|\KP{1.2} c\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\le
\sup\kern0.37mm\big\{\,|\KP1\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|:
\eta\in{}^{}\Cal Omega\KP1\} } holds for $\varphi\in\smb\Phii\in S\ar 1$ and all
$\eta\in{}^{}\Cal Omega\,$. Then by density of $S\ar 1$ and completeness of
$\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm
(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm)$ letting $c$ be the unique continuous
extension of $c\ar 1$ we first get
$c\in
\Cal L\,(\kern0.37mm G_{\kern0.15mm/\kern0.37mm S}\kern0.37mm,\kern0.15mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm
(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm)) \,$, and further using classical
convergence results for sequences of measurable functions we see that also
$c$ is a choice function.
\end{proof} | 2,866 | 362,273 | en |
train | 0.47.67 | For a positive measure \math{\mu} on \math{{}^{}\Cal Omega} and for \mathss38{ X =
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\kern-0.3mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.15mm) }, \,by a {\it lift\kern0.37mm} of a linear
subspace \math{S} in \math{X} one means a linear map \math{c:X_{\kern0.37mm|\,S} \to
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.15mm{}^{}\Cal Omega\kern0.15mm) } that is also a choice function such
that for \math{(\kern0.37mm\smb\Phii,\kern0.07mm\varphi\kern0.37mm)\in c} and for every \math{
\varphi\ar 1\in\smb\Phii } and \math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+} we \linebreak
have \mathss38{ \sup\KP1(\kern0.37mm\varphi\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\kern0.37mm) \le \sup\KP1(\kern0.37mm
\varphi\ar 1\kern-0.3mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\kern0.37mm) }. So from the proof of
Proposition \ref{Pro lift} we see that we could have more specifically stated
that a lift exists. However, below we shall have no essential use of this
additional information encoded in the definition of lift. \vskip.3mm
Essentially the content of \cite[Theorem 8.17.2\kern0.37mm, p.\ 582]{Edw} is in the
following
\begin{proposition}\label{Pro Edw 8.17.2}
Let $\,\mu$ be an almost decomposable positive measure on $\,{}^{}\Cal Omega${\,\rm, }
and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{LCS}\kern0.4mmps0(K)$ be normable and
such that $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable topology. Also let \ú$\,\smb U \in
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm))$ be such
that there is a choice function \ú$\, c \in
\Cal L\,(\kern0.37mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)
_{\kern0.37mm/\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\kern-0.2mm\sixmath U} \, , \kern0.37mm
\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm)) \, $. Then there
is \ú$\, y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with $\, \smb U =
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\capss31
\{\,(\kern0.37mm\xi\,,\kern0.15mm\smb\Phii\kern0.07mm) : \roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y \in
\smb\Phii\,\} \KP1 $.
\end{proposition}
\begin{proof} With \math{ y = \seqss33{
\roman{ev}\kern0.15mm\sbi\eta\kern-0.2mm\circ\kern0.15mm c\circss11\smb U:\eta\in{}^{}\Cal Omega} } we have \math{
y} a function \mathss38{{}^{}\Cal Omega\to\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) }, \,and
for fixed \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} noting that \math{c} is a choice function, we
obtain \inskipline{.5}2
(\kern0.15mm$*$\kern0.15mm) $ \KP{10.55}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y
= \seqss33{\roman{ev}\kern0.15mm\sbi\eta\kern-0.2mm\circ\kern0.15mm c\circss11\smb U\fvalss10\xi :
\eta\in{}^{}\Cal Omega}
= \seqss33{c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\smb U\fvalss10\xi\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta:\eta\in{}^{}\Cal Omega}$ \inskipline{.5}{29}
${}
= c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\smb U\fvalss10\xi\kern0.37mm)
\in \smb U\fvalss10\xi
\in \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) \KP1 $. \KP6 From
this, \inskipline{.5}0
we see that \math{\smb U =
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\capss31
\{\,(\kern0.37mm\xi\,,\kern0.15mm\smb\Phii\kern0.07mm) : \roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y \in
\smb\Phii\,\} } holds.
It remains to verify that \math{y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } holds. First, to
prove that $(\KPt5 y\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$
is finitely almost scalarly measurable, let
$\scrmt A$ and $N\kern0.15mmrim1$ be as in
Definitions \ref{df decomp}\,(2) on page \pageref{decos A} above, and let
$D$ be countable and $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\,$--\,dense.
Then for every fixed
$A\in\scrmt A$ and $\xi\in D$ from (\kern0.15mm$*$\kern0.15mm) we see existence of
$N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
with
\linebreak
\ú$B=A\kern0.15mm\setminus N\kern0.37mm$ we have
\math{(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\KP1|\KP1 B\,;
\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } measurable.
By countability of $D$ we can here take $N$ independent of
\mathss34{\xi\in D}.
Then by the {\sl axiom of choice\kern0.15mm}
in conjunction with the decomposability property we get
existence of a $\mu\,$--\,negligible $N\kern0.15mmrimm1$ such that
for all $A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ and $\xi\in D$
with $B=A\kern0.15mm\setminus N\kern0.15mmrimm1$ we have
$(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\KP1|\KP1 B\,;
\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm)$
\noindent
almost measurable.
By countability and
density of $D$ we then see that
$(\KPt5 y\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$
is finitely almost scalarly measurable. | 3,547 | 362,273 | en |
train | 0.47.68 | ${}
= c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\smb U\fvalss10\xi\kern0.37mm)
\in \smb U\fvalss10\xi
\in \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) \KP1 $. \KP6 From
this, \inskipline{.5}0
we see that \math{\smb U =
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\capss31
\{\,(\kern0.37mm\xi\,,\kern0.15mm\smb\Phii\kern0.07mm) : \roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y \in
\smb\Phii\,\} } holds.
It remains to verify that \math{y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } holds. First, to
prove that $(\KPt5 y\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$
is finitely almost scalarly measurable, let
$\scrmt A$ and $N\kern0.15mmrim1$ be as in
Definitions \ref{df decomp}\,(2) on page \pageref{decos A} above, and let
$D$ be countable and $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\,$--\,dense.
Then for every fixed
$A\in\scrmt A$ and $\xi\in D$ from (\kern0.15mm$*$\kern0.15mm) we see existence of
$N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
with
\linebreak
\ú$B=A\kern0.15mm\setminus N\kern0.37mm$ we have
\math{(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\KP1|\KP1 B\,;
\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } measurable.
By countability of $D$ we can here take $N$ independent of
\mathss34{\xi\in D}.
Then by the {\sl axiom of choice\kern0.15mm}
in conjunction with the decomposability property we get
existence of a $\mu\,$--\,negligible $N\kern0.15mmrimm1$ such that
for all $A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ and $\xi\in D$
with $B=A\kern0.15mm\setminus N\kern0.15mmrimm1$ we have
$(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\KP1|\KP1 B\,;
\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm)$
\noindent
almost measurable.
By countability and
density of $D$ we then see that
$(\KPt5 y\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$
is finitely almost scalarly measurable.
To complete the proof of \mathss38{y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) }, \,by \math{
\bouSet(\kern0.15mm\vPi\dlbetss01\kern0.15mm)\subseteq\bouSet(\kern0.15mm\vPi\dlsigss00\kern0.07mm)} it suffices
to show that \math{{}^{}{\rm rng}\,{}_{{}^{}} y\in\bouSet(\kern0.15mm\vPi\dlbetss01\kern0.15mm) } holds. For this,
we first note that there is some \mathss03{\smb A\in\lbb R_+} such that for \math{
\varphi\in\smb\Phii\in{}^{}{\rm rng}\,{}_{{}^{}}\smb U} we have \mathss36{
\|\KP1 c\fvalss11\smb\Phii\,\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu \le \smb A \KP1
\|\KP1\varphi\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu}. Then with \math{ B\ar 1 =
{}^{}{\rm rng}\,{}_{{}^{}}\smb U\capss21\{\KPt8\smb\Phii \kern-0.2mm : \aall{\varphi\in\smb\Phii}\,
\|\KP1\varphi\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu\le 1\KPt8\} } taking \math{ U =
\smb U\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm B\ar 1 } we have \mathss03{U\in\neiBoo\vPi } and
hence \vskip.3mm\centerline{$
|\KPp1.1 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi\KP1|
= |\KPp1.1 c\circss00\smb U\fvalss11\xi\fvalss10\eta\KP1|
\le \|\KP1 c\circss00\smb U\fvalss11\xi\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu
\le \smb A\KP1\|\KP1\varphi\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu
\le \smb A $} \inskipline{.5}0
for \math{\eta\in{}^{}\Cal Omega} and \math{\xi\in U} and \mathss35{ \varphi \in
\smb U\fvalss11\xi}. Consequently \math{ {}^{}{\rm rng}\,{}_{{}^{}} y \in
\bouSet(\kern0.15mm\vPi\dlbetss01\kern0.15mm) } holds.
\end{proof} | 2,469 | 362,273 | en |
train | 0.47.69 | The essential content of \cite[Theorem 8.17.6\kern0.37mm, p.\ 584]{Edw} is in the
following
\begin{proposition}\label{Pro Edw 8.17.6}
Let $\,\mu$ be an almost decomposable positive measure on $\,{}^{}\Cal Omega${\,\rm, }
and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{LCS}\kern0.4mmps0(K)$ be normable and
such that $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable topology. Then for every \ú$\,\smb V
\in\Cal L\,(\kern0.37mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$
there is \ú$\, y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ such that $\,
\smb V\fvalss60\smb\Phii\fvalss00\xi = \int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu$ holds for $\,
\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V$ and $\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm$.
\end{proposition}
\begin{proof} We first get a continuous bilinear map \math{ \beta :
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\kern0.37mm\sqcap\kern0.15mm\vPi\to\bosy K } defined by \mathss35{
(\kern0.37mm\smb\Phii\kern0.07mm,\kern0.07mm\xi\kern0.37mm)\mapsto\smb V\fvalss70\smb\Phii\fvalss01\xi }, \,
and then a continuous linear map \math{ \smb U : \vPi \to
\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } de- fined by \math{\xi
\mapsto\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\beta\,(\,\cdot\,,\kern0.07mm\xi\kern0.37mm)) }
where \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1} is as in Proposition \ref{Pro L^1'=L^i} on page \pageref{Pro L^1'=L^i}
above. Then by Propositions \ref{Pro lift} and \ref{Pro Edw 8.17.2} there is \math{
y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } with \vskip.4mm\centerline{$
\smb U =
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\capss31
\{\,(\kern0.37mm\xi\,,\kern0.15mm\smb\Psii\kern0.15mm) : \roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y \in
\smb\Psii\,\} \KP1 $.} \inskipline{.4}0
Now for \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} with \math{\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V }
and \math{\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y \in
\smb\Psii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } we
have \mathss03{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm\smb V\,) =
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\beta\,(\,\cdot\,,\kern0.07mm\xi\kern0.37mm)) =
\smb U\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\xi = \smb\Psii } and hence \math{
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm\smb V =
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.37mm\smb\Psii } whence finally \mathss36{
\smb V\fvalss60\smb\Phii\fvalss00\xi =
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm\smb V\,\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii =
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.37mm\smb\Psii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii =
\int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu}.
\end{proof} | 2,571 | 362,273 | en |
train | 0.47.70 | The content of \cite[Lemma 8.17.8 \erm A\kern0.37mm, p.\ 584]{Edw} is in the
following
\begin{lemma}\label{Le 8.17.8 A}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\roman{LCS}\kern0.4mmps0(K)$ be normable{\kern0.37mm\rm, }and
let \ú$\,F=E\dlbetss12\,$. Also let $\,S\ar 1$ be a closed linear subspace in $\,
F$ such that \ú$\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 S\ar 1$ is a separable topology. Then there
is a closed linear subspace $\,S$ in $\,E$ with \ú$\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 S$ a
separable topology and such that \ú$\,\seq{\KP1 u\KPt9|\KP1 S:u\in S\ar 1\,}$
is a strict morphism \ú$\, F_{\kern0.15mm/\kern0.37mm\aars S_1} \to
(\kern0.15mm E_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12 $ in the sense of
{\,\rm\cite[Definition 2.5.1\kern0.15mm, p.\ 100]{Ho}\,.}
\end{lemma}
\begin{proof} Fixing a compatible norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \mathss35{E}, \,let \math{
\bosy u\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,S\ar 1} be such that ${}^{}{\rm rng}\,{}_{{}^{}}\bosy u$ is \mathss34{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 S\ar 1}--\,dense, and let $\bosy x\in\kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb24 I)$ be such that $
(\kern0.37mm 1 + i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm\big)\KP1
(\kern0.37mm\bosy u\,.\KPt8\bosy x\fvalss01 i\kern0.37mm)
=\sup\kern0.37mm\big\{\KPt8|\KP1\bosy u\fvalss01 i\fvalss10 x\KP1|:
x\in\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb23 I\,\}$ holds for all $i\in\mathbb No\,$. Then we let
$S$ be the closed linear span of ${}^{}{\rm rng}\,{}_{{}^{}}\bosy x$ in $E\,$, and take \mathss38{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm=\seqss40{u\KPt9|\KP1 S:u\in S\ar 1} }.
One easily verifies that \math{
\sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb23 I\,\}
\le \sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy x\KPt9\} } holds
for every fixed $u\in S\ar 1\kern0.37mm$, and hence we get \vskip.6mm
$
\sup\kern0.37mm\big\{\KPt8|\KP1\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss10 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb20 I\capss32 S\KP1\}
= \sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb20 I\capss32 S\KP1\}
$ \inskipline{.4}{10.6}
${}\le \sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb23 I\,\}
\le \sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy x\KPt9\}
$ \inskipline{.4}{10.6}
${}\le \sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb20 I\capss32 S\KP1\}
= \sup\kern0.37mm\big\{\KPt8|\KP1\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss10 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb20 I\capss32 S\KP1\} \KP1 $, \inskipline{.6}0
from which the assertion easily follows.
\end{proof} | 2,759 | 362,273 | en |
train | 0.47.71 | The content of \cite[8.17.8\kern0.37mm, pp.\ 584\,--\,586]{Edw} is in the following
\begin{proposition}\label{Pro Edw 8.17.8}
Let $\,\mu$ be an almost decomposable positive measure on $\,{}^{}\Cal Omega${\,\rm, }
and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ and \ú$\,\smb V\in
\Cal L\,(\kern0.37mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$
be such that $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.07mm\vPi^{\kern0.4mm\prime}_{\kern-.2mm\raise.95mm\hbox{$_{_\beta}$}}\kern0.07mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32{}^{}{\rm rng}\,{}_{{}^{}}\smb V$ is a
separable topology. Then there is \ú$\, y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with $\,
{}^{}{\rm rng}\,{}_{{}^{}} y\subseteq\CltaurdvPidualbeta{}^{}{\rm rng}\,{}_{{}^{}}\smb V$ and such that \vskip.1mm\centerline{$
\smb V\fvalss60\smb\Phii\fvalss00\xi = \int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu $} \inskipline{.5}0
holds for $\,\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V$ and $\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm$.
\end{proposition}
\begin{proof} Taking \math{S\ar 1=\CltaurdvPidualbeta{}^{}{\rm rng}\,{}_{{}^{}}\smb V } and \math{E=
\vPi} in Lemma \ref{Le 8.17.8 A} above, there is a closed linear subspace \math{
S} in \math{\vPi} with \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 S } a separable topology and
such that for \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm=\seq{\KP1 u\KPt9|\KP1 S:u\in S\ar 1\,} } we have \math{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} a strict morphism \mathss37{\vPi\dlbetss01{}_{\kern0.15mm/\kern0.37mm\aars S_1} \to
(\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12 }.
Now we have \mathss38{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\circ\kern0.15mm\smb V \in
\Cal L\,(\kern0.37mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) \, , \kern0.15mm
(\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12\kern0.15mm) }, \,and by separability of the
topol- \linebreak
ogy \math{ \tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm) =
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 S } we can apply Proposition \ref{Pro Edw 8.17.6} to
deduce existence of some \linebreak
$y\ar 1 \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm(\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlsigss12) \kern0.37mm$
with \inskipline{.5}2
(\kern0.15mm r\kern0.15mm) $ \KP{12}
\smb V\fvalss60\smb\Phii\fvalss00\xi
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\smb V\fvalss60\smb\Phii\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\xi
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\circ\kern0.07mm\smb V\fvalss60\smb\Phii\fvalss00\xi = \int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\kern-0.63mm\cdot\varphi\rmdss21\mu$ \inskipline{.5}0
whenever \math{\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V} and \math{\xi\in S} hold.
Then taking \math{ y = \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\inve\kern-0.2mm\circ\kern0.15mm y\ar 1 } we get a function \mathss38{
y:{}^{}\Cal Omega\to S\ar 1\subseteq\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) }, \,and it remains
to establish \ú$\kern0.37mm y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) $ and to show that \inskipline{.5}2
(\kern0.15mm s\kern0.15mm) $ \KP{12}
\smb V\fvalss60\smb\Phii\fvalss00\xi = \int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu$ \inskipline{.5}0
holds for \math{\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V} and \mathss31{ \xi \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi}.
Noting that \math{\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlsigss12) \subseteq
\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12\kern0.15mm) } by Banach\,--\,Steinhaus,
we see that \vskip.5mm\centerline{$
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlsigss12) \subseteq
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12\kern0.15mm) \subseteq
\bouSet(\kern0.15mm\vPi\dlbetss01{}_{\kern0.15mm/\kern0.37mm\aars S_1}\kern0.07mm\sbig)0 \subseteq
\bouSet(\kern0.15mm\vPi\dlbetss01\kern0.15mm) \subseteq \bouSet(\kern0.15mm\vPi\dlsigss12) \KP1 $,} \inskipline{.5}0
and hence \math{y} is similarly \q{finitely almost bounded} as \math{y\ar 1}
is. So, in order to establish \linebreak
\ú$y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) \kern0.37mm$ we only need
to show that \math{(\KPt5 y\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } is finitely
almost scalarly measurable. For this, arbitrarily fixing \mathss31{ \xi \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }, \,we first observe that \linebreak
$\roman{ev}\kern0.15mm\sbi\xi\,|\KP1 S\ar 1 \in
\Cal L\,(\kern0.15mm\vPi\dlbetss01{}_{\kern0.15mm/\kern0.37mm\aars S_1},\kern0.07mm\bosy K\kern0.37mm) \kern0.37mm$ and
hence also \mathss38{
\roman{ev}\kern0.15mm\sbi\xi\,|\KP1 S\ar 1\kern-0.2mm\circ\kern0.07mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\inve \in
\Cal L\,(
(\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12{}_{\kern0.37mm/\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\kern-0.2mm\emath\iotaa}\kern0.37mm , \kern0.07mm
\bosy K\kern0.37mm) }. | 3,841 | 362,273 | en |
train | 0.47.72 | (\kern0.15mm s\kern0.15mm) $ \KP{12}
\smb V\fvalss60\smb\Phii\fvalss00\xi = \int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu$ \inskipline{.5}0
holds for \math{\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V} and \mathss31{ \xi \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi}.
Noting that \math{\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlsigss12) \subseteq
\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12\kern0.15mm) } by Banach\,--\,Steinhaus,
we see that \vskip.5mm\centerline{$
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlsigss12) \subseteq
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12\kern0.15mm) \subseteq
\bouSet(\kern0.15mm\vPi\dlbetss01{}_{\kern0.15mm/\kern0.37mm\aars S_1}\kern0.07mm\sbig)0 \subseteq
\bouSet(\kern0.15mm\vPi\dlbetss01\kern0.15mm) \subseteq \bouSet(\kern0.15mm\vPi\dlsigss12) \KP1 $,} \inskipline{.5}0
and hence \math{y} is similarly \q{finitely almost bounded} as \math{y\ar 1}
is. So, in order to establish \linebreak
\ú$y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) \kern0.37mm$ we only need
to show that \math{(\KPt5 y\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } is finitely
almost scalarly measurable. For this, arbitrarily fixing \mathss31{ \xi \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }, \,we first observe that \linebreak
$\roman{ev}\kern0.15mm\sbi\xi\,|\KP1 S\ar 1 \in
\Cal L\,(\kern0.15mm\vPi\dlbetss01{}_{\kern0.15mm/\kern0.37mm\aars S_1},\kern0.07mm\bosy K\kern0.37mm) \kern0.37mm$ and
hence also \mathss38{
\roman{ev}\kern0.15mm\sbi\xi\,|\KP1 S\ar 1\kern-0.2mm\circ\kern0.07mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\inve \in
\Cal L\,(
(\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12{}_{\kern0.37mm/\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\kern-0.2mm\emath\iotaa}\kern0.37mm , \kern0.07mm
\bosy K\kern0.37mm) }.
By separability of the topology \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12\kern0.15mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm } we are
able to apply Lemma \ref{Le 8.17.8 B} on page \pageref{Le 8.17.8 B} above to
get existence of \math{ \bosy\xi \in \kern0.15mm ^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,S } with \math{{}^{}{\rm rng}\,{}_{{}^{}}\bosy\xi
\in\bouSet\vPi } and \vskip.4mm\centerline{$
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\inve\fvalss02\zeta\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\xi
= \roman{ev}\kern0.15mm\sbi\xi\,|\KP1 S\ar 1\kern-0.2mm\circ\kern0.07mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\inve\fvalss02\zeta\ar 1
=\lim\,(\kern0.37mm\zeta\ar 1\kern-0.3mm\circ\kern0.37mm\bosy\xi\kern0.37mm) $} \inskipline{.4}0
for all \math{\zeta\ar 1\in{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm }, \,and hence \math{\zeta\fvalss11\xi =
\lim\,(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss11\zeta\circss11\bosy\xi\kern0.37mm) } for all \mathss34{
\zeta\in S\ar 1}. In particular for \linebreak
\ú$(\kern0.37mm\eta\kern0.37mm,\kern0.07mm\zeta\kern0.37mm)\in y \kern0.37mm $
we obtain \mathss38{\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta =
\lim\,(\kern0.37mm y\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\circss11\bosy\xi\kern0.37mm) }, \,giving the
required measurability. To get (\kern0.15mm s\kern0.15mm) we note that by the above we also
have \inskipline{.7}{8.5}
$ \smb V\fvalss60\smb\Phii\fvalss00\xi
= \lim\,(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\circ\kern0.07mm\smb V\fvalss60\smb\Phii\circ\kern0.15mm\bosy\xi\kern0.37mm)
= \lim\,(\kern0.37mm\smb V\fvalss60\smb\Phii\circ\kern0.15mm\bosy\xi\kern0.37mm)$ \inskipline{.5}{19}
${}
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}\kern0.15mm\int_{\KP{1.1}{}^{}\Cal Omega\,}
y\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm)\KP1(\kern0.37mm
\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)
= \int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu \,$, \inskipline{.7}0
where we used (\kern0.15mm r\kern0.15mm) with dominated convergence, noting that it is
legitimate by the above established boundedness and measurability properties.
\end{proof} | 2,983 | 362,273 | en |