I know an example for finite lebesgue measure for an open dense set, but not for zero measure. Otherwise can we prove that no such set exists?
2026-04-29 19:15:48.1777490148
Does there exist an open dense subset of $\mathbb{R}$ with Lebesgue measure zero?
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No. Any open subset $U\subseteq \mathbb{R}$, dense or not, contains a small interval $(x_0-\varepsilon,x_0+\varepsilon)$ and therefore has positive Lebesgue measure.