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> Let $A$ be a Lebesgue measurable subset of $\Bbb R$. 
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> 1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^\*(B\setminus A)=0$. 
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> 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.
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> (Note that $l^\*$ denote the outer measure. )
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> 

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.