Show the following measure theory result.

Question

Let $A$ be a unbounded Lebesgue measurable subset of $\mathbb{R}$ such that $m(A) < \infty$. Show that for each $\epsilon > 0$,there exists a bounded Lebesgue measurable set $B$ in $\mathbb{R}$ such that $B \subset A$ and $m(A \setminus B)< \epsilon $.
I have no idea how to approach these kind of problems.

Answer 1

By Monotone Convergence Theorem, we have $m(A\cap[-n,n])\rightarrow m(A)$. Now $m(A)<\infty$ allows us to have the estimate that $m(A)-m(A\cap[-n,n])<\epsilon$ for large $n$.