Convex hulls of families of probability measures

Question

Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$.

In this paper for any family of probability measures $P\subset \mathcal P(X)$ its strong convex hull is defined as $$ \operatorname{sco}P:=\left\{\left.\int_{\mathcal P(X)}q\;\nu(\mathrm dq)\;\right|\;\nu\in \mathcal P(\mathcal P(X)): \nu^*(P) = 1\right\} \subset \mathcal P(X). $$ which is exactly the set of all convex combinations of elements of $P$: here $\nu^*(P)$ is the outer $\nu$-measure of the set $P$. Since $\operatorname{sco}$ is a monotone map, I guess it admits at least one fixpoint which can be perhaps referred to as a convex closure of $P$. I'm not sure though whether the fixpoint is unique.

My question concerns the literature on convex hulls and convex closures of families of measures e.g. on Borel spaces. The aforementioned paper only contains a single reference to "Probability and Potential" by Dellacherie and Meyer, however I have only volume C available in my library, and there I did not find by any means a comprehensive study of these concepts. Any hint is greatly appreciated.

In particular, besides the uniqueness of the fixpoint I am interested whether the limit $$ \operatorname{clo} P := \bigcup_{n\in \Bbb N}\operatorname{sco}^n P $$ is a fixpoint of $\operatorname{sco}$.

Answer 1

This is more a comment but I don't have that privilege. If I understand your question correctly, then two very comprehensive papers on the topologies of spaces of probability measures by T. Banakh should be of interest. These originally appeared in russian but are now easily available in english translation (arxiv: 1112.6161 and 1206.1727).

Answer 2

See the notion of "measure-convex set" ...

MR2269765 Dostál, Petr; Lukeš, Jaroslav; Spurný, Jiří Measure convex and measure extremal sets. Canad. Math. Bull. 49 (2006), no. 4, 536–548.

MR1009196 Rosenthal, Haskell Martingale proofs of a general integral representation theorem. Analysis at Urbana, Vol. II (Urbana, IL, 1986–1987), 294–356, London Math. Soc. Lecture Note Ser., 138, Cambridge Univ. Press, Cambridge, 1989.

also see MO post Integral in a σ−convex set.

Answer 3

Many thanks to Gerald Edgar and D. Kelleher for their answers. I am (and was) a little familiar with the Choquet's theory and tried to find the answer to OP and similar questions there (mostly in "Lecture notes on Choquet's theorem"), but my search was not very successful. By no means it implies that the answer can't be found there - just perhaps my familiarity with the Choquet's theory is insufficient.

Inspired by the fact that the paper by Dostal et al. (Gerald's first reference) mentions a concept of a barycentre which also appears in the book "Probabilities and Potential, C" by Dellacherie and Meyer (English version, 1988), I've decided to look one more time into this book with focus on this keyword. There the Theorem in Chapter XI, par. 33 (p. 196) shows that for all strongly convex sets $P$ it holds that $\operatorname{sco}P = P$ and for all analytic sets $P$ the set $\operatorname{sco}P$ is analytic and strongly convex (in fact, it is the strongly convex envelope of $P$ being and intersection of all strongly convex supersets of $P$). The authors put a remark that the result is "most probably due to Fremlin", and Dostal et al. also contain a reference to Fremlin's work so perhaps for the origins I'll have to look into that direction.

In particular, we have now that $\operatorname{sco}$ maps the class of analytic subsets of $\mathcal P(X)$ into itself, and it is idempotent on this class. Together with the comments by D. Kelleher it may give a nice characterizaion of fixed points of $\operatorname{sco}$.