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 $\operatorname{sco}$.