Given a set $X$ and a topology $\tau$ on $X$ the definition of the Borel $\sigma$-algebra $B(X)$ makes use of the availability of open sets in the topological space $(X, \tau)$: it is the $\sigma$-algebra generated by the open sets. There are many ways to generalize the notion of a topology, e.g.
(i) preclosure spaces (with a closure operator that is not necessarily idempotent) or equivalently
(i') neighborhood system spaces (a neighborhood of a point need not contain an "open neighborhood") and more generally
(ii) filter or net convergence spaces (satisfying some convergence axioms).
The notion of convergence spaces is strong enough to be able to speak of continuity of maps (defined by preservation of convergence). If $X$ is a convergence space then one can form the set $C(X)$ of continuous real-valued functions $f : X \to \mathbb{R}$ (where $\mathbb{R}$ is equipped with the convergence structure coming from its usual topology). In this way, one can at least relate such spaces to measure theory by creating the Baire $\sigma$-algebra $Ba(X)$ on $X$ generated by $C(X)$.
Questions:
Are there other ways to connect such generalized topological structures to measure theory and probability theory on such spaces that are of interest in practice? I especially may think here of applications in functional analysis where Beattie and Butzmann argue that convergence structures are more convenient than topologies in such a context. As a standard example, the notion of almost everywhere convergence (of sequences, nets or filters) is not topological.
Are there some practical applications in working with such Baire $\sigma$-algebras in non-topological preclosure or convergence spaces? Even for topological space, the Baire $\sigma$-algebra and the Borel $\sigma$-algebra need not coincide. (I think they do coincide if $\tau$ is perfectly normal).
Is the following only a trivial idea or does it lead to interesting properties: To any convergence space one can assign a topological space (the reflection of the convergence space, see ncatlab) and thus speak of a "Borel" $\sigma$-algebra for a convergence space.
I also understand that measure theory on general topological spaces can be rather boring. Only for special topological spaces like Polish spaces or Radon spaces we may have interesting measure-theoretic results. So maybe there is also an interesting class of non-topological convergence spaces with interesting measure-theoretic theorems generalizing those for Radon spaces.