What does the statement (Let $K$ be the choquet simplex of all probability measures on $X$) mean?

Question

Let $X$ be a probability measure space.

What does the statement (Let $K$ be the choquet simplex of all probability measures on $X$) mean ?

It is mentioned in C. Lance. Ergodic Theorems for Convex Sets and Operator Algebras page 202 the second and third lines.

Answer 1

It's mostly a name.

"Choquet" because of the analogy with the case where you consider the state space of a function system $K\subset C(X)$ for a compact Hausdorff space $X$ ($K$ is a unital selfadjoint subspace). As the dual of $C(X)$ are the regular Borel measures, the state space of $K$ consists of the probability measures; and one calls it "simplex" because it is the closed span of its extreme points, namely the point measures.

When $X$ is just a set, you can still consider the probability measures, and still the extreme points will be the point measures. So "Choquet simplex". But you shouldn't read much on it.

Answer 2

In a fairly general setting (LCTVS), a point in a compact convex set K can be represented as the barycenter of a probability measure on the extreme points of K. When K is a Choquet simplex, this representation is unique. In the case at hand, the phrase draws attention to the fact that an invariant measure is canonically a direct integral of ergodic invariant measures.