If $E \subset \mathbb{R}$ is measurable with $m(E) > 0$, must it contain a closed interval? I know it has to contain a closed set $F$ with $m(E \setminus F) < \epsilon$ (for any $\epsilon$), but I don't know if it must contain a closed interval.
If $E \subset \mathbb{R}$ is measurable with m(E) > 0, must it contain a closed interval?
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real-analysis
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0@StevenTaschuk No, and neither does the empty set :P – 2012-03-16
2 Answers
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No. For example, $\mathbb{R} \setminus \mathbb{Q}$ has infinite measure but contains no interval.
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No. Take the irrational numbers, for example.
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0@M Turgeon The two answers have the same time stamp! – 2013-09-13