Just having trouble with this problem. First, it says to prove that if $E$ is Lebesgue Measurable, and $\epsilon>0$ is arbitrary, then there is an open $O$ such that $E \subset O$ and $m(O\setminus E)<\epsilon$. Now for this part, since $E$ is Lebesgue measurable, I was able to take the definition of being Lebesgue outer measurable (the infimum of open coverings of $E$) and easily construct $O$ from that, and it works out.
But now I want to show that there is an $F$ closed with $F \subset E$ and $M(E\setminus F)<\epsilon$. The problem is, I can't figure out how to construct this $F$! The idea seems obvious intuitively, but I feel like I am given no tools for constructing a closed subset of an arbitrary set that satisfies these conditions. Am I missing something obvious here?