Let $A\subseteq [0,1]$ and suppose for each $\epsilon>0$, exists a measurable set $B$ such that $m^*(A\triangle B)<\epsilon/2$. Show that $A$ is measurable.
So my try was using the fact that for $B\setminus A \subseteq\Bbb R$, there exists an open set $G$ such that $m^*(B\setminus A)\leq m^*(G)\leq m^*(B\setminus A)+\epsilon/2$, and then choosing a set $O=A\cup G$, then I get:
$m^*(O\setminus B)=m^*((A\cup G)\setminus B)\leq m^*(A\setminus B)+m^*(G\setminus B)\leq m^*(A\triangle B)+m^*(G)\leq \epsilon$
But I don't know if this works, because I don't really know if $O$ is an open set in order for the Lebesgue measure definition to work.