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 $\lt F_n(x) \gt$ converges almost everywhere iff $\lt F_n(x) \gt$ 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 $\lt F_n(x) \gt$ be a sequence of measurable functions on a measure space X. Then $\lt F_n(x) \gt$ converges in measure iff every subsequence of $\lt F_n(x) \gt$ 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:

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

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

3. 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.)