How Do You Prove the Equivalence of These Definitions of Measurability?

[AxiomOfChoice]

One possible definition of measurability is this: A set $$E \subseteq \mathbb R^d$$ is (Lebesgue) measurable if for every $$\epsilon > 0$$ there exists an open set $$\mathcal O \supseteq E$$ such that $$m_*(\mathcal O \setminus E) < \epsilon$$. Here, $$m_*$$ indicates Lebesgue outer measure.

Apparently, an equivalent definition is this: "For every $$\epsilon > 0$$ there exists a closed set $$F \subseteq E$$ such that $$m_*(E\setminus F) < \epsilon$$."

Showing the equivalence of these definitions was a practice problem recently for the final exam in my real analysis class. But I couldn't get it, and even though I'm on break now, it's bugging me. Can someone help? Thanks! (This is also apparently a problem in Stein-Shakarchi's textbook, Real Analysis.)

 

[quasar987]

==>: Suppose E is measurable. Then E^c is measurable. Let O be the open set associated to E^c as in the definition of measurability. Then use F=O^c.

<==: Same thing, just use the open/closed duality in the same way.