A well known classical fact is that a Lebesgue measurable function on Euclidean space is almost everywhere equal to a Baire class 2 function. A relatively modern reference for this fact is van Rooij - Schikhof, "A second course on real functions", Cambridge, 1982, p. 143. See also the Math Overflow question here.

What happens in more general situations, say for a measurable function $f$ with respect to a finite regular measure $\mu$ on a locally compact space $X$? Does this have to be equal ae ($\mu$) to some Baire class 2 function on $X$? I think it is true for $X$ compact, because you can use Lusin's theorem to get zero sets $C$ of almost full measure on which $f$ restricted to $C$ is continuous. (Does anyone know of a reference for this?) But for the general locally compact case I think it might not be true.