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Let $M(X)$ be the vector space (actually it's an algebra) of all equivalent classes of measurable functions $X\to \mathbb{C}$ (where $X$ is a measured space) modulo equality almost-everywhere.

One can define a notion of sequential convergence on $M(X)$. A sequence $(f_n)$ of $M(X)$ converges to $f$ iff (by definition) there is a sequence of functions $g_n$ and a function $g$ such that $g_n\to g$ pointwise almost everywhere and where $g_n$ is in the class of $f_n$ and $g$ is in the class of $f$. This notion of convergence is compatible in an obvious way with the algebraic structure of $M(X)$.

From what I understood of what I read, $M(X)$ cannot be given a topology such that the notion of convergence associated with this topology coincide with the previous notion of convergence almost everywhere.

My question is about a positive result : what is the structure of almost everywhere convergence ? In other words, what is $M(X)$ ? Is it a "convergence vector space" (whatever that would mean) ? (From this, it is not clear to me that this is the case)

Has this structure been explored, for example, could we talk about completeness in this setting, how about classical topological results such as Hahn-Banach ? Are they still true in some form for this "convergence space" structure ?

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Yes, this defines a "convergence vector space". In fact, it's probably the original motivating example for the generalization. In Fréchet's thesis he discussed L-spaces, which are essentially sequential convergence spaces: a set equipped with families of convergent sequences at each point satisfying the axioms that the constant sequence converges and subsequences of convergent sequences converge. The axioms of a convergence structure are intended to generalize this by axiomatizing nonsequential convergence. However, given an L-space you can define a convergence space generated by the tail filters $\{ x_n : n \geq m \}$ of the convergent sequences. Fréchet also discussed the special case of E-spaces, which we call metric spaces, and this special case became much more popular.

With messier notation than the definitions in your link, you could define a convergence structure in terms of families of convergent nets such that the constant net converges and subnets of convergent nets converge (given the appropriate notation of subnet that corresponds to a subfilter), but since there is no set of all nets on a space it's easier to just define convergent filters.

Much of functional analysis generalizes to convergence vector structures in some way. Perhaps the most salient new feature of the theory is an improved duality theory, with continuous convergence providing a dual object corresponding to aw$^*$ (almost weak$^*$) topology on locally convex spaces, which often fails to even be a topology, nevermind a linear topology.

The Hahn-Banach Theorem doesn't generalize outright, as it often fails quite badly for topological vector spaces, existence of spaces with no nontrivial continuous linear functionals. However, the question of when the Hahn-Banach Theorem holds for a convergence vector space is an interesting one, even for the duals of locally convex spaces, where the convergence vector space dual of $E$ having the Hahn-Banach extension property is equivalent to $E$ being fully complete (or B-complete) in the ordinary sense of locally convex spaces. This is connected to the generalizations of the Closed Graph Theorem for locally convex spaces, and gives better proofs of those results.

The best book on the subject is Convergence Structures and Applications to Functional Analysis by Beattie and Butzmann. It's a pretty intriguing theory, but unfortunately many of the consequences for the theory of locally convex spaces had already been discovered in different forms with less clear statements and proofs.

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