Given the notion of some kind of convergence of sequences or nets on a set, one question is whether there exists a topology for that convergence and find it. The last section of Chapter 2 in Kelley's General Topology provides one characterization.
Since sigma algebra and topology are similar in many aspects, I was wondering if there is a concept for a sigma algebra, and/or from which people ask whether there exists a sigma algebra for a given collection of "convergent" objects and find it? Note that the concept needs not be a mimic of convergence for topology.
Although the concept may not be a mimic of convergence, I think to define something like convergence for a sigma algebra $\mathcal{F}$ on $\Omega$, one possibility is to
first define a "net" or "sequence" by considering a measurable mapping from a directed set $D$ or $\mathbb{N}$ with its discrete sigma algebra to the underlying set $\Omega$ of the sigma algebra $\mathcal{F}$, and
then define "convergence" of a "net" or "sequence" as $x \to \infty$ in $D$ or $\mathbb{N}$, and
then one can study what properties of the collection of all "convergent" "nets" or "sequences" has, and if they can in turn characterize the sigma algebra $\mathcal{F}$ from a given collection of "nets" or "sequences" such that the "nets" or "sequences" "converge". What do you think?
Thanks and regards!
By the way, is this kind of questions okay at MO?