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:
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Are there other natural notions of convergence which don't exactly correspond to convergence in some topology?
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Are there other pairs of natural convergences which have a similar topological relationship as convergence in measure and convergence ae?
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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.)
Best Answer
This is a well established concept in General Topology: «convergence structures». The two references I would recommend are the first chapters of each one of the following books:
E. Binz, Continuous Convergence on $C(X)$. LNM Springer, 469.
R. Beattie and H.-P. Butzmann, Convergence Structures and Applications to Functional Analysis.
A quick overview: On a set $X$, for every $x\in X$ we define which filters converge to $x$, with the following restrictions: the ultrafilter of all supersets of $x$ must converge to $x$; any filter which contains a filter converging to $x$ must converge to $x$; the intersection of two filters converging to $x$ must converge to $x$ («contains» and «intersection» to be understood in the usual set-theoretic sense).
Converging filters in a convergence subspace: a filter ${\mathcal F}$ converges to $x$ in the subspace iff the filter on the initial space generated by the filter basis ${\mathcal F}$ converges to $x$.
The sets which are present in every filter converging to $x$ are known as neighborhoods of $x$ with respect to the corresponding convergence structure $\Lambda$. Such sets are actually neighborhoods in the topological sense, for a topology on $X$ (called the topology associated with $\Lambda$). The definitions of a closed set as a set whose complementary is open, and as a set which coincides with its (filterwise) adherence, are equivalent. A set $K$ is said to be $\Lambda$-compact if every ultrafilter in $K$ converges with respect to the induced convergence structure on $K$. Filterwise continuous maps send compact sets onto compact sets.