Give a counterexample or prove this statement 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?

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.
