Undergrad Help understanding a passage from a proof of change of variables formula

[psie]

TL;DR

I'm reading a proof of the change of variables formula for a $C^1$ diffeomorphism $G:\Omega\subset\mathbb R^n\to\mathbb R^n$ in Folland for the Lebesgue integral, that is $$\int_{G(\Omega)}f(x)\,dx=\int_\Omega f\circ G(x)|\det D_xG|\,dx.$$Here $D_xG$ is the Jacobian matrix. There's a reference to an earlier theorem that I struggle with.

Here's an excerpt from the proof of the change of variables formula in Folland's book (Theorem 2.47, page 76, 2nd edition, 6th and later printings):

Let $W_K=\Omega\cap\{x:|x|<K\text{ and } |\det D_xG|<K\}$. If $E$ is a Borel subset of $W_K$, by Theorem 2.40 there is a decreasing sequence of open sets $U_j\subset W_{K+1}$ such that $E\subset \bigcap_1^\infty U_j$ and $m\left(\bigcap_1^\infty U_j\setminus E\right)=0$.

For reference, see Theorem 2.40 below. I don't understand how he is using Theorem 2.40 in the quoted passage. Which part of Theorem 2.40 is he using? Moreover, I don't understand the role of $W_K$ and in particular, why $U_j\subset W_{K+1}$? It'd be awesome if someone could clarify these points.

 

[psie]

I've worked it out I think. He uses part (a) and everything follows from this. I know, I didn't give a lot of context, but the proof is long.