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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.

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    Does a singleton count as a closed interval?2012-03-16
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    @StevenTaschuk No, and neither does the empty set :P2012-03-16

2 Answers 2

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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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    I guess that, to be fair, one should give +1 to your answer if and only if one gives +1 to Chris' answer!2012-03-16
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    @M Turgeon The two answers have the same time stamp!2013-09-13