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 ?