Dirac Delta Measure I have a question about the Dirac delta measure. Does the "sifting" property hold at the limits of integration? That is, does $\int_{[a,b]}f(x)\delta_b(dx)=f(b)$?
Thanks...

Yes, so long as you're sure to include the endpoint $b$. The Dirac delta measure at $b$ is defined by 
$$\delta_b (A) = \chi_A (b) := \begin{cases}
1 & \textrm{if } b \in A \\
0 & \textrm{otherwise.}
\end{cases}
$$
The integral of $f$ over the region $[a,b]$ is just the integral of $f \chi_{[a,b]}$; so we are interested in the measure $\chi_{[a,b]}\delta_b$:
$$
\chi_{[a,b]}\delta_b (A) = \int \chi_A \chi_{[a,b]}\delta_b 
= \int \chi_{A \cap [a,b]}\delta_b 
= \delta_b(A \cap [a,b])=\delta_b(A)
$$
since $b\in[a,b]$ implies $b \in A \cap [a,b] \iff b \in A$. 
Thus we have
$$\int_{[a,b]} f \delta_b = \int f \chi_{[a,b]} \delta_b = \int f \delta_b = f(b).$$
This reflects the fact that from the point of view of measure theory and Lebesgue integration, the endpoint isn't special in any way - we can define the Dirac delta measure for any point of any measure space.
