TITLE: Notions of convergence not corresponding to topologies
QUESTION [61 upvotes]: 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.)

REPLY [4 votes]: Regarding your point 3 here is a list of some sources you might find interesting:

Dolecki & Mynard, "Convergence Foundations of Topology": topology from convergence theory point of view, basically this is what you are asking about
Schechter, "Hadbook of Analysis and its Foundations": good book on analysis that uses nets and filters in an essential way
Nel, "Continuity Theory"
Dolecki, "An invitation into Convergence Theory"

You can find more of them if you'll look for "convergence spaces" as this is a name of the object of the theory you asked for. Also you can look for "filters" and "nets".
As a side remark I can mention that there's a course on basic analysis that introduces limits using filter convergence (kind of): Zorich "Mathematical Analysis".
Also note, that the category of convergence spaces is quite general and there are some useful subcategories of it that are still more general than the topological spaces: pretopological and pseudotopological spaces.