[Math] About measurable sets and intervals

Question

Given a Lebesgue measurable set A with strictly positive measure, can we find an open interval (a,b) such that x belongs to A for almost every x in (a,b)?

Thanks in advance for any comments!

Answer 1

The usual Cantor set constructed by removing 1/3 at each step is nowhere dense but has measure 0. However, there exist nowhere dense sets which have positive measure. The trick is to try to remove less, for instance you remove 1/4 from each side of [0,1] during the first step then 1/16 from each pieces etc...

The resulting set is the fat Cantor set: it is nowhere dense and it has positive measure.