What does it mean for a function to be invertible 'almost everywhere'

Question

It seems to me that the correct definition of a measure-theoretic inverse for a function f is a function g such that $f \circ g$ and $f \circ g$ are the identity almost everywhere. The problem I have with this definition is that this doesn't seem to identify itself with a category theoretic inverse (if we identify measurable functions equal almost everywhere, this doesn't form a quotient category). Furthermore, I do not believe measure preserving transformations invertible almost everywhere need not have an inverse which is measure preserving. What are some alternate/'correct' definitions of an 'almost everywhere inverse'.