Question:

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

and

. Thus, let

and

<--Suppose for any

there are lebesgue measurable sets

such that

and that

.

Let

* and *
Clearly,

and A and B are both measurable.

And

for all n; thus,

.

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