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Let $m^*$ be an Lebesgue outer measure on a set $X$. Show that a subset $E$ of $X$ is $m^*$-measurable if and only if for each $\epsilon>0$ there exist a $m^*$-measurable set $F$ such that $F$ is a subset of $E$ and $m^*(E\setminus F)<\epsilon$.

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What have you tried? Do you know how to accept an answer? People will be much more likely to answer if you show some effort in approaching your own question, and if, checking your profile, they will find that you are used to accept good answers when proposed. –  Giovanni De Gaetano Oct 18 '12 at 13:22
In specific I would say that one of the two directions of the "if and only if" is obvious. Cannot you prove at least one of the two? –  Giovanni De Gaetano Oct 18 '12 at 13:25
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