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Let $E \subseteq \mathbb{R}$ s.t. $\mu^*(E) < \infty$ for some outer measure $\mu^*$ on $\mathcal{P(\mathbb{R})}$. Must there exist an open set $O$ s.t. $O \supseteq E$ and $\mu(O) < \infty$? What if we let $\mu$ and $\mu^*$ be the Lebesgue Outer Measure and Lebesgue Measure respectively?

I'm reading a proof which seems to assume that, at least the case of the Lebesgue setting, there must exist such an $O$. But I'm thinking that if we let $E = \mathbb{N}$ then $m^*(E) = 0 < \infty$ yet I can't think of an open set $O$ that contains $\mathbb{N}$ with finite outer measure so I'm confused.

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    What is your definition of Lebesgue outer measure? $m^*(E)$ is the infimum of the sums of lengths of intervals covering $E$. The intervals can be taken to be open.2012-12-03

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