Give a counterexample or prove this statement

Question

We all know that in Real Analysis, if $E$ is Lebesgue measurable, then there is a $F_{\sigma}$ set $F$ such that: $$F\subset E,\ m(E\backslash F)=0$$

But I want to know whether the next statement hold:

Suppose $E$ is a Lebesgue measurable set, is there a $F_{\sigma}$ set $F$ such that: $$E\subset F,\ m(F\backslash E)=0 $$

Personally, I don't think this statement is correct. But I found it hard to give a counterexample, can anyone help me?

Answer 1

For a counterexample, let $E$ be a dense $G_\delta$ set of measure zero. Assume for a contradiction that $F$ is an $F_\sigma$ set of measure zero such that $E\subseteq F$. Then $G=\mathbb R\setminus F$ is a dense $G_\delta$ set. But then $E$ and $G$ are two disjoint dense $G_\delta$ sets, contradicting the Baire category theorem.

Answer 2

The general answer to your question is NO. For a measurable set $E$ what we can to is this: $$ A \subseteq E \subseteq B,\qquad m(E\setminus A)=0,\qquad m(B\setminus E)=0 $$ where $A$ is an $F_\sigma$ set and $B$ is a $G_\delta$ set.