Two results in classical measure theory (Q1813496)

We prove that if \(A\) is a Lebesgue measurable subset of the real line having positive Lebesgue measure, then there exists a closed interval \(I\) such that \(D(A\cap I)\) equals \(D(I)\), where \(D(C)\) is the set of all real numbers \(c-c'\) with \(c\in C\) and \(c'\in C\). We also prove that there exists a subset \(B\) of the real line such that \(B\) has positive outer Lebesgue measure and is of the second Baire category and such that \(B\) contains no subset that is symmetric about a point and at the same time either has positive outer measure or is of the second Baire category.