Suppose that $E \subset \mathbb{R}$. If $E \cap F$ is Lebesgue measurable for all measurable $F$ such that $m(F)$ is finite, then $E$ is measurable.

Question

Suppose that $E \subset \mathbb{R}$. If $E \cap F$ is Lebesgue measurable for all measurable $F$ such that $m(F)$ is finite, then $E$ is measurable.

I have spent hours trying to prove this statement. Any hints? Note: $m(F)$ is the Lebesgue measure.