If $S$ and $T$ with $S \subset T$ are 2 Lebesgue measurable sets, does there exist an open set $O$ with $S \subset O \subset T$?
An open set in between 2 measurable sets
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real-analysis
2 Answers
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Not necessarily. For example, $S=\{0\}$ and $T=\{0,1\}$ are Lebesgue measurable subsets of $\mathbb{R}$, and $S\subset T$, but there is no open set $O$ of $\mathbb{R}$ such that $S\subset O\subset T$.
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2Lol. We usually think about barbarian examples with irrationals and stuff ; this one is just gold. – 2012-02-14
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A counterexample even exists with both sets having positive measure. For example the irrationals in $(0,1/2)$ and the irrationals in $(0,1)$.