Let $A$ be a Lebesgue measurable subset of $\Bbb R$.
1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$.
2) Show that every Lebesgue measurable set is the union of a Borel measurable set (with same measure) and a set of Lebesgue measure zero.
(Note that $l^*$ denote the outer measure. )
I already showed that if $C$ is a Lebesgue measurable subset of $\Bbb R$ and $\epsilon \gt 0$, then there exists an open set $G_{\epsilon}\supseteq C$ such that $l^*(C)\le l^*(G_{\epsilon})\le l^*(C)+\epsilon$.
Moreover, if $D$ is a Lebesgue measurable subset of $\Bbb R$, if $\epsilon \gt 0$, and if $D\subseteq I_n=(n,n+1]$, then there exists a compact set $K_{\epsilon} \subseteq D$ such that $l^*(K_{\epsilon})\le l^*(D)\le l^*(K_{\epsilon})+\epsilon$.
My thought is to somehow manipulate the inequality and have $\epsilon$ shrinking, but I'm stuck to show this and come up with the result. Could someone help to provide a proof please? Any help is appreciated. Thanks.