Given $E\subset\mathbb{R}^n$ and for every $\epsilon>0$ there are $F,G\subset\mathbb{R}^n$ $F\subset E\subset G$ and $V(G\backslash F)<\epsilon$.

Question

Given $E\subset\mathbb{R}^n$ and for every $\epsilon>0$ there are $F,G\subset\mathbb{R}^n$ jordan measureable $F\subset E\subset G$ and $V(G\backslash F)<\epsilon$.
Prove E is jordan measureable.

My try:We can see that because E is a subset of G we can deduce that E is bounded.
I assume from the given information that $G\backslash F$ is measure zero set and got stuck here..

Answer 1

For any $\epsilon > 0$ if $F,G$ are as described, then by their measurability there must be a cover $\mathscr G$ of $G$ by cells with $V(\mathscr G) - V(G) < \epsilon$ and a collection of cells $\mathscr F$ contained in $F$ with $V(F) - V(\mathscr F) < \epsilon$. So, how big can $V(\mathscr G) - V(\mathscr F)$ be?

Note that $\bigcup\mathscr F \subseteq E \subseteq \bigcup\mathscr G$.