Take the 2-minute tour ×  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.
share|improve this question Did you mean to answer someone's
question? –  David Roberts Oct 29 '12 at 6:44 My initial post
was an accident –  Fred Dashiell Oct 29 '12 at 7:0