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Assume we are working in the reals under the standard Lebesgue measure.

Must a Borel set B of nonzero measure contain an interval as a subset?

I conjecture yes. Is the following line of reasoning valid? Borel sets are generated by countable intersections, countable unions, and complements of open sets, but all sets of nonzero measure must be a union of uncountably many points. The union may contain an interval or be composed entirely of discrete points (each with some neighborhood containing no other point from the union). Since Borel sets only allow for the mentioned countable operations and B has nonzero measure, the nonzero measure cannot be due to an uncountable union of discrete points, so the nonzero measure must be due to some interval contained in B.

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The set $\mathbb{R} \setminus \mathbb{Q}$ or all irrational numbers is a G$_\delta$-set of full/infinite Lebesgue measure (or is a co-null set if, you prefer) which includes no intervals.

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This fat Cantor set is a counterexample: closed, empty interior, yet positive Lebesgue measure.

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  • $\begingroup$ Wow, I think you are right. That's pretty cool. $\endgroup$ – Just Some Old Man Mar 26 '13 at 1:52
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    $\begingroup$ I wonder what was wrong with my reasoning. Countable unions (and intersections) of countable sets are countable. However, countably many countable unions of countable sets may be uncountable. I am not sure, but I think this may be what was false with my reasoning. $\endgroup$ – Just Some Old Man Mar 26 '13 at 1:56
  • $\begingroup$ Wait, a countable union of countable set is countable. So a countable union of countable unions of countable sets is countable. Since $\mathbb{N}\times \mathbb{N}\times\mathbb{N}\simeq \mathbb{N}\times\mathbb{N}\simeq\mathbb{N}$. $\endgroup$ – Julien Mar 26 '13 at 2:01
  • $\begingroup$ To begin with: I don't understand your alternative: "the union may contain an interval or be composed of discrete points". $\endgroup$ – Julien Mar 26 '13 at 2:12
  • $\begingroup$ You are certainly correct about countable iterations of countable unions being countable. As for my alternative, an uncountable union of points can either contain an interval as a subset, or not. My mistake, which you correctly pointed out, was that the latter case is not equivalent to each point containing a neighborhood such that no other point lies in that neighborhood. In retrospect, my interpretation was quite ridiculous since it ignores the whole concept of a limit point. For example, any neighborhood about a rational must contain another rational. And the rationals are 'just' countable! $\endgroup$ – Just Some Old Man Mar 26 '13 at 2:29

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