I have read that if $A$ is Lebesgue-measurable, then there exists Borel sets $B,C$, with $B\subset A\subset C$, such that $m(B) = m(C) = m(A)$. It is clear for me that such a set C exists, just by taking intersections of unions of rectangles covering $A$. But I don't see how to obtain the set $B$.
Thank you!