Let $P$ be a set of positive Lebesgue measure in $\mathbb{R}^n$ and $O$ be an open set in $\mathbb{R}^n$ such that $E=P\cap O$ is a set of zero Lebesgue measure. Can we conclude that $P\setminus \overline{E}$ is a set of positive Lebesgue measure?
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We cannot, it may even be empty. Try constructing such a situation starting from an appropriate $E$. – Daniel Fischer Jul 25 '20 at 14:34
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Do need to consider $P$ to be a fat Cantor set? – Mathemajician Jul 25 '20 at 14:45
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No. Let $n=1$. Let $O$ be an open set which contains the rationals and has finite measure. Let $P=(\mathbb{R}\setminus O)\cup\mathbb{Q}$. Then $E=\mathbb{Q}$ so $P\setminus \overline{E}=\emptyset$.
halrankard
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See here: https://math.stackexchange.com/questions/201410/open-measurable-sets-containing-all-rational-numbers – halrankard Jul 25 '20 at 14:52
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