I am trying to prove the following problem:
If $A\subset \mathbb R$ such that $A\cap B$ is Lebesgue measurable for every bounded subset $B$ of $\mathbb R$, then $A$ is Lebesgue measurable.
If $A$ itself if bounded, then the problem is solved. But what to do if $A$ is unbounded. Any hint will be appreciated.
Countable unions of measurable sets are measurable; using your final line, one can prove that for each integer $n$,
$$A_n := A \cap [n, n + 1)$$
is measurable. Finally,
$$A = \bigcup_{n \in \mathbb{Z}} A_n$$