# $F_\sigma$ subset of equal outer measure

Let $E\subset[a,b]$.

How can we show that:

$\exists F \in F_\sigma$: $F\subseteq E$ and $\mu^*(F)=\mu^*(E)$

Can we conclude that $E$ is measurable?

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I think $E$ must be measurable since there is a non-measurable set $H$ in $[a,b]$ such that for any measurable subset $A$ of $H$,we have $μ(A)=μ^*(A)=0$ but $μ^*(H)>0$.
WLOG,let $a=0,b=1$.Define an equivalence relation ~ on $[0,1]$ s.t. $a$~$b \iff a-b \in \mathbb{Q}$.Let $H$ be the representatives of each equivalence class.Then $[0,1] \subseteq \bigcup_{q\in [0,1] \cap\mathbb{Q}}(H+q)\subseteq [-1,2]$.It can shown that $H$ is not measurable and for every measurable subset $A$ of $H$,we have $μ(A)=0$.However,$μ^*(H)>0$,otherwise $H$ is measurable since $μ$ is complete measurable.