# Measure theory and Probability Theory on generalizations of topological spaces

I think this MSE thread is more suitable for the MO community, so I copy it here.

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" of that point) and more generally

(ii) filter convergence spaces or certain net convergence spaces (e.g. Fréchet $L$-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:

1. Are there other known 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 (at least from a category theoretic point of view). As a standard example, the notion of almost everywhere convergence is not topological.

2. Are there some practical applications in working with such Baire $\sigma$-algebras in non-topological preclosure or convergence spaces? Even for topological spaces, the Baire $\sigma$-algebra and the Borel $\sigma$-algebra need not coincide. (I think they do coincide if $\tau$ is perfectly normal).

3. 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 the "associated 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.

• Have you looked at Bogachev massive book on measure theory? – Liviu Nicolaescu Aug 16 '16 at 10:18
• @Liviu Nicolaescu: Yes, Bogachev only considers measures on topological spaces (if you mean his two-volume book on measure theory). – yada Aug 16 '16 at 11:00
• Do you have a concrete example in mind where the existing theory is not applicable? – Liviu Nicolaescu Aug 16 '16 at 13:04
• @Liviu Nicolaescu: Your question is related to my question, namely whether measure theory on certain non-topological convergence spaces was already investigated by someone - I wasn't able to find anything in the literature so far. Some obstacle may be that usually a generalization of some theory looses structure, e.g. in convergence spaces we still can speak of compactness or separatedness (Hausdorffness) but I think something like "perfect normality" has no direct generalization since we are not able to speak of open and closed sets anymore. – yada Aug 16 '16 at 13:24
• @LiviuNicolaescu: For any question which Bogachev's massive book doesn't answer, the next place to turn might be Fremlin's even more massive book. – Nate Eldredge Aug 20 '16 at 0:59

The measures that are approximated from within by such compact-like classes have been studied, with interesting non-trivial results. A good entry point to this area is Fremlin's paper "Weakly $\alpha$-favourable measure spaces", Fund. Math. 165 (2000), 67--94.