Question: $\displaystyle A_{\epsilon} \leq E \leq B_{\epsilon} $ makes no sense since they are sets. Do you mean they are subsets or their measures are less than or equal? It looks like you mean subsets, so I'll use that, but the proof is nearly the same either way.

-->Suppose E is Measurable

If E is measurable, then there exists a G-delta set, G and a F-sigma set, F such that $\displaystyle F \subseteq E \subseteq G$ and $\displaystyle m(G \setminus F) = 0 $. Thus, let $\displaystyle A_{\epsilon} = F $ and $\displaystyle B_{\epsilon} = G $

<--Suppose for any

there are lebesgue measurable sets

such that

and that $\displaystyle m(B_{\epsilon} \setminus A_{\epsilon})<\epsilon $.

Let $\displaystyle A = \bigcap_{n=1}^{\infty}A_n $

* and *$\displaystyle B = \bigcap_{n=1}^{\infty}B_n$

Clearly, $\displaystyle A \subseteq E \subseteq B $ and A and B are both measurable.

And $\displaystyle m(B \setminus A) \leq 1/n $ for all n; thus, $\displaystyle m(B \setminus A) = 0 $.

So E differs by A (or B) by a zero set, so E is measurable.