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

Question

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.

Answer 1

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.