# If $E \subset \mathbb{R}$ is measurable with $m(E) > 0$, must it contain a closed interval?

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? –  Steven Taschuk Mar 16 '12 at 6:58
@StevenTaschuk No, and neither does the empty set :P –  badatmath Mar 16 '12 at 20:09
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## 2 Answers

No. For example, $\mathbb{R} \setminus \mathbb{Q}$ has infinite measure but contains no interval.

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I see, thank you. –  badatmath Mar 16 '12 at 1:14
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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! –  M Turgeon Mar 16 '12 at 1:17
@M Turgeon The two answers have the same time stamp! –  Stefan Smith Sep 13 '13 at 23:57
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