Let E a set contained in R with non-empty interior.
Then i know that it can be seen as the countable union of open disjoint intervals and every intervals has positive measure. So, there exist a subset of E s.t. is not Lebesgue measurable (Vitali-set like).
But if E has EMPTY INTERIOR?
I think that, due to the fact that Lebesgue measure has no atom, necessarily the measure of E is zero, so there is no non-measurable subset. Is it correct my idea?
Best Answer
Every set of positive measure in $\mathbb R$ contains a non-measurable subset. [See https://math.stackexchange.com/questions/2079436/does-every-non-null-lebesgue-measurable-set-contain-a-non-measurable-subset?rq=1 ]. If $C$ is fat Cantor set then $C$ has no interior but there is a non-measurable subset.