Question about Folland's proof of the Change of Variables

Question

In his proof, Folland shows that if $G$ is a diffeomorphism on $\Omega\subseteq \mathbb{R}^n$, and $U$ is an open subset of $\Omega$, then $$m(G(U))\le \int_U|\det D_xG|$$ To establish the same inequality for a Borel measurable set, he starts with an arbitrary Borel set $E\subseteq \Omega$ of finite measure and then uses a decreasing sequence of open subsets $(U_k)^\infty_{k=1}$, each of finite measure, which include $E$, with $m(\bigcap_k U_k)=m(E)$, to assert that $$m(G(E))\le m(G(\bigcap_k U_k))=\lim_k m(G(U_k))\le\lim\int_{U_k}|\det D_xG|=\int_E|\det D_xG|$$ where the equality of limits is justified by the Dominated Convergence theorem. But there is no justification for the finiteness of either $m(G(U_k))$ or the integral of $\int_{U_k}|\det D_x G|$ when $m(U_k)<\infty$ and thats why I'm puzzled.

Answer 1

Ok, its an error in his exposition. I found the correction at

https://sites.math.washington.edu/~folland/Homepage/oldreals.pdf