MathOverflow Asked by yada on December 10, 2020
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:
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.
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).
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.
For question 1: There is another way to generalize the notion of a topology, different from (i), (i') and (ii): Extract an abstract notion of a compact-like class of sets.
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.
Correct answer by user95282 on December 10, 2020
This answer is an addendum to the one of user95282. There is a class of spaces which are based on compacta and for which a sophisticated theory of measures has been developed. These are the compactological spaces which were introduced by H. Buchwalter based on work by L. Waelbroeck. They are basically the objects of Grothendieck’s ind construction applied to the category of compact spaces. One advantage of this approach is that it allows perfect analogues of the classical Riesz and Gelfand Naimark dualities for compacta to the non-compact situation. A series of articles by various authors, with detailed expositions of a range of topics, is now readily available online (in the journal “Publ. Dépt. Math. Lyon” at the site EuDML.org). Particularly relevant to your question is “Espaces de mesures et compactologies” in vol. 9 (1972), 1-35 by J. Berruyer and B. Ivol. The seeding article by Buchwalter is “Topologies et Compactologies” in vol. 6 (1969), (2) 1-74.
Answered by user131781 on December 10, 2020
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