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

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.

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.
• Wait, a countable union of countable set is countable. So a countable union of countable unions of countable sets is countable. Since $\mathbb{N}\times \mathbb{N}\times\mathbb{N}\simeq \mathbb{N}\times\mathbb{N}\simeq\mathbb{N}$. – Julien Mar 26 '13 at 2:01