Cardinality of Lebesgue measurable functions under a.e.-identification

Question

Does anyone know the cardinality of $$ \{f\colon \mathbb{R}\to\mathbb{R} \text{ Lebesgue measurable}\}/\sim $$ where $$ f\sim g\Leftrightarrow \text{f=g Lebesgue a.e.}? $$

Answer 1

Yes: it's $2^{\aleph_0}$. A Lebesgue-measurable function is a.e. identical to a Borel-measurable function. The set of Borel-measurable functions has cardinality $2^{\aleph_0}$. Of course, since all constants are Lebesgue-measurable, there are at least $2^{\aleph_0}$ pairwise non-almost-everywhere-identical measurable functions.