Proof- verification : About change of variable on Lebesgue integration.

Question

I'm proving the change of variable formula on $\mathbb{R}$.

Suppose $F$ is absolutely continuous increasing funciton on $[a,b]$ and assume $m(O)=\int_{F^{-1}\left(O\right)}F'(x)\ dx$ holds for all open set $O\subset[a,b]$. Then $$ m(G)=\int_{F^{-1}\left(G\right)}F'(x)\ dx $$ holds for all $G$ : $G_{\delta}$-set.

This is a partial lemma from Integration by substitution for Lebesgue integration. @copper.hat relied that using Dominated convergence theorem, this is obvious. But for disjoint open sets $O_{1}$ and $O_{2}$, above statement holds : $m(O_{1}\cup O_{2})=m(O_{1})+m(O_{2})$, and $F^{-1}\left(O_{1}\cup O_{2}\right)=F^{-1}\left(O_{1}\right)\cup F^{-1}\left(O_{2}\right)$. But $G_{\delta}$-set is countable union of open sets which do not have to be 'disjoint'. How can I see this? Thanks in advance.

Answer 1

Since $G$ is a $G_{\delta}$, there is a $decreasing$ sequence $\left \{ O_n \right \}$ of open sets such that $\cap O_n=G$ which implies that $m(G)=\lim m(O_n)$.

Now, $m(O_n)=\int_{F^{-1}\left(O_n\right)}F'(x)\ dx=\int _{[a,b]}\chi_{F^{-1}(O_n)}(x)\cdot F'(x)dx$,

so that, noting that $\chi_{F^{-1}(O_n)}\to \chi_{F^{-1}(G)}$ as $n\to \infty$, and that $\vert F'(x)\vert $ is bounded a.e. on $[a,b]$, an application of the $DCT$ gives

$\lim m(O_n)=m(G)=\lim \int _{[a,b]}\chi_{F^{-1}(O_n)}(x)\cdot F'(x)dx=\int _{[a,b]}\lim (\chi_{F^{-1}(O_n)}(x)\cdot F'(x))dx=\int _{[a,b]}\chi_{F^{-1}(G)}(x)\cdot F'(x)dx=\int _{F^{-1}(G)}F'(x)dx.$