Question about the positive variation of a signed measure If $(X,\Sigma)$ is a measurable space and $v$ is a signed measure, I want to prove that $$v^+(E)=\sup\{v(F):F\subseteq E,F\in\Sigma\}$$
where $E\in\Sigma$.
Let $X=P\cup N$ a Hanh descomposition for $v$. So we know that $v^+(E)=v(E\cap P)$, and it's clear that $v^+(E)\le\sup\{v(F):F\subseteq E,F\in\Sigma\}$.
But I don't know how to prove the inequality $v^+(E)\ge\sup\{v(F):F\subseteq E,F\in\Sigma\}$.
Can anyone give my a hint? Thank you.

Let $$E^+ = \sup\{\nu(F) : F\subset E, F\in\Sigma\}.$$ If $F\in\Sigma$, $F\subset E$ then $$\nu(F) = \nu^+(F) - \nu^-(F)\leqslant \nu^+(E)-\nu^-(F)\leqslant \nu^+(E),$$ so that $\nu^+(E)\geqslant E^+$.
