This question concerns the ramifications of the following interesting problem that appeared on Ed Nelson's final exam on Functional Analysis some years ago:

Exam question: Is there a metric on the measurable functions on R such that a sequence $\langle F_n(x) \rangle$ converges almost everywhere iff $\langle F_n(x) \rangle$ converges in the metric?

Answer: No.

Better Answer: Convergence ae does not even precisely correspond to a topology!

The later answer follows from the following (textbook) theorem:

Theorem: Let $\langle F_n(x) \rangle$ be a sequence of measurable functions on a measure space X. Then $\langle F_n(x) \rangle$ converges in measure iff every subsequence of $\langle F_n(x) \rangle$ has a subsequence converging almost everywhere.

In particular:

Corollary: If one places a topology T on the measurable functions such that all the almost-everywhere convergence sequences converge in T, then all the convergent-in-measure sequences also converge in T.

Obvious questions are:

Are there other natural notions of convergence which don't exactly correspond to convergence in some topology?

Are there other pairs of natural convergences which have a similar topological relationship as convergence in measure and convergence ae?

Can one construct a nice theory of "convergences" different from the theory of topologies? (Warning: This problem tortured me for some weeks some years ago.)