Convergence of Measurable Sets

Question

Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of measurable subsets of $\mathbb{R}$. Suppose $A \subset \mathbb{R}$ such that $A_n \subset A$ for each $n \in \mathbb{N}$ and $m^{*}(A \setminus A_n) \rightarrow 0 $. Show that $A$ is lebesgue measurable.

My attempt : I know that $m^{*}(A \setminus A_n) \rightarrow 0$ $\;$ for large enough $n$, $\;$$m^{*}(A \setminus A_n) = 0$ $\;$ so $A \setminus A_n$ is measurable. From here i don't know how to proceed. Any help would be much appreciated!

Answer 1

Let $B=\cup_n A_n$ Then $B \subset A$ and $B$ is measurable. Also $m^{*}(A\setminus B) \leq m^{*}(A\setminus A_n)$ for all $n$ so $m^{*}(A\setminus B)=0$. It follows that $A=(A\setminus B) \cup B$ is measurable.