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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.

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    For the second part, write $f(x)=R(x)+iI(x)$, where $R(x)$ is the real part and $I(x)$ is the imaginary part. Now apply Lusin's theorem to both functions separately and find a way to make the result work for both simultaneously. – 2017-01-22

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