Lusin's Theorem for finite measure spaces

Lusin's Theorem. Let $f$ be a real-valued measurable function on $E$. Then for each $\varepsilon>0$, there is a continuous function $g$ on $\mathbb{R}$ and a closed set $F$ contained in $E$ for which $f=g$ on $F$ and $m(E\sim F)<\varepsilon$.

can anybody guide me the proof of the theorem or can advice me some book from where i can go through the proof.