TITLE: Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?
QUESTION [156 upvotes]: I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability space rather than Lebesgue measurable functions. This is so in every textbook on probability theory which I consulted. In general, it seems to me that probability theory favors the Borel algebra more than the algebra of Lebesgue measurable sets. My question is: why?
In every course in measure theory, one is taught of the notion of a complete measure and completion of measures and I got the impression that a complete measure space is somewhat superior to a non-complete one (or at least that completeness makes life a bit easier on the technical level), so this preference of Borel sets puzzles me.

REPLY [9 votes]: On pages 209-210 of his book "Real Analysis, A Comprehensive Course in Analysis, Part I" Barry Simon gives an argument for sticking with Borel measurable sets and functions.
Edit:
Barry Simon argues that Lebesgue measurable functions are not closed under composition, that it complicates arguments such as constructing product measures, requiring an extra completion set, and that nothing is gained since every Lebesgue measurable function is equal a.e. to a Borel function, and equivalence classes that matter.