If $A \subset B \subset \mathbb{R}$ such that $A$ is Lebesgue measurable and $m^*(B)=m(A)< \infty$. Show that $B$ is Lebesgue measurable.

Question

If $A \subset B \subset \mathbb{R}$ such that $A$ is Lebesgue measurable and $m^*(B)=m(A)< \infty$. Show that $B$ is Lebesgue measurable.

I am doing this by showing that there exist a closed and open sets that are subset and superset of given set respectively. But I can't show the superset part.How to do this?

Answer 1

Since $A$ is measurable and $A \subset B$, we have $m^*(B-A) = m^*(B) - m^*(A)=0$. Therefore, $B-A$ is measurable.

Since $B = A \cup (B-A)$, it follows that $B$ is measurable.