This is an exercise in real analysis:

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> Let $E\subset {\Bbb R}^d$. For every $\varepsilon>0$, one can find a Lebesgue measurable set $E\_{\varepsilon}$ such that $m^\*(E\_{\varepsilon}\Delta E)\leq\varepsilon$. Show that $E$ is Lebesgue measurable.
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The definition of Lebesgue measurable used here is 

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> Let $E\subset{\Bbb R}^d$, $E$ is Lebesgue measurable if for any $\varepsilon>0$, there exists an open set $U\supset E$ such that $m^\*(U\setminus E)\leq\varepsilon$, where $m^\*$ is Lebesgue outer measure. 
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Directly using the definition above might be difficult. I'm trying to use the fact that the Lebesgue measurable sets form a $\sigma$-algebra. But I don't see the way to write $E$ as a union of Lebesgue measurable set. Any help?