I read up on "Convergence space" on Wikipedia and An initiation into convergence theory - Szymon Dolecki. I think I get the general idea. However, I study within applied math, so I am trying to see how this very general concept translates to more specific concepts.
First, for example, as far as I understand, convergence space does generalize convergence in measure - but how? How do "measure" and the sequence of functions translate to the families of subsets of convergence space?
Second, other examples would be series that converge or dynamical systems. Maybe an example here would help me to understand how to apply the idea of convergence space.
Third, as part of my research, I think about processes (basically a sequence of states) that converge to a state. The states in such a process get progressively better approximations of the state the process is converging to. Here, I say that there are states in time $s_t\in p$ in the process $p$ after which no state becomes a worse approximation than the state $s_t$. I tried to map this to convergence space, because I do not have notion of "approximation" really, except that I can talk about sets of states -- there is no definition of "similarity" between states. While I can see that could say
Let $F_t$ be a filter such that the set {$s_{t'}$}${}_{t'\ge t}$ is an element of $F_t$.
and then try to build a convergence from those $F_t$, I understand that for a relation to be a convergence, all filters defined on the set of states need to adhere to the convergence axioms; and, I do not see whether that is making sense in my case.