There exists no topology on the set of Lebesgue-measurable functions such that a sequence is convergent in that topology if and only if it is convergent almost everywhere (i.e. everywhere except for a set of Lebesgue measure zero). That's because if all the sequences which converge almost everywhere are convergent in some topology, then all the sequences which are convergent in measure will also convergent in that topology?
My question is, does there exist a topology on the set of measurable functions such that a net is convergent in that topology if and only if it is convergent almost everywhere? Or does nothing change if you replace sequences with nets?