I know that Lebesgue measure is completion of Borel measure and there is zero Lebesgue measured set that is not Borel measurable. However I could not be sure that is there any set with positive Lebesgue measure which is not Borel measurable.
I appreciate for any explanation or suggest etc.
Best Answer
Just take a non-Borel measurable null set and take a disjoint union with a set of positive Lebesgue measure. The result will have positive Lebesgue measure and not be Borel measurable.