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I've been studying for a test in analysis and I became stuck on this problem. Exactly stated it is:

Prove or disprove: If $U$ is a subset of $\mathbb{R}^n$ whose boundary has outer measure zero, then $U$ is Lebesgue measurable.

Most of my attempts so far have been to try to find a subset of $\mathbb{R}^n$ which is Lebesgue measurable, but with a boundary constructed in a Vitali non-measurable set sort of way, in order to disprove the statement. However, this has been fruitless. Now I'm leaning towards the statement being true, but still have no idea how to use outer measure to demonstrate it. Any help would be appreciated.

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$U$ is the union of its interior (which is open, therefore measurable) and a subset of its boundary (which has outer measure $0$, therefore is Lebesgue measurable).

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