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.