Let $X$ be a polish space (separable completely metrizable topological space). Let $m$ be a probability measure on $X$ and $f:X \rightarrow \mathbb{R}$ a measureable function. I want to show that $f$ is equal a.e. to a second Baire class function (limits of limits of bounded continuous functions).

I know how to prove this when $f$ is bounded: in that case $f$ is in $L_1(X)$ and there exists a sequence of continuous functions which converge to $f$ a.e. and are uniformly bounded. Then by taking some limits I can deal with the points in which the sequence doesn't converge.

My question is, how to prove for any function $f$, not necessarily bounded?

Thanks, Shlomi