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The question is simple:

Does every Borel set with a positive Lebesgue measure contain a closed interval $[a,b]$ with $a<b$?

If not than I need a counterexample; if so some kind of proof would be nice. I have no idea how to get closed intervals into a Borel set.

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    $\begingroup$ The irrationals? $\endgroup$ May 7, 2017 at 17:50
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    $\begingroup$ Positive-measure Cantor sets? $\endgroup$ May 7, 2017 at 17:59

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As pointed out by Zestylemonzi, the set of irrational numbers has an infinite measure and does not contain any closed or non-empty open interval because any such interval contains rational numbers.

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