Lusin's Theorem: Let $f$ be a real-valued measurable function on $E$. Then for each $\epsilon > 0$, there is a continuous function $g$ on $\Bbb R$ and a closed set $F$ contained in $E$ for which $f =g$ on $F$ and $m(E - F)< \epsilon$.
Some extensions of Lusin's Theorem:
a) Prove the extension of Lusin's Theorem to the case that E has infinite measure.
b) Prove the extension of Lusin's Theorem to the case that f is not necessarily real-valued, but may be finite a.e. (almost everywhere).
I have proved the first part but struggling with the second part. Help Needed.