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.
