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Suppose $A \subseteq \mathbb{R}^n$ is an open set. Can we conclude that $A$ does not have measure zero??

I am trying to find an open set with measure zero, but it seems quite hard to construct one such a set. Can someone help me construct such a set ? Maybe the result is true, and there is not an open set of measure zero.

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up vote 6 down vote accepted

Hint: A non-empty open subset of the real line contains an interval. Similarly, a non-empty open subset of $\mathbb{R}^n$ contains a ball.

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Non-empty is crucial here. There is exactly one open set of measure zero. – user2357112 Apr 3 '14 at 9:13

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