In Walters' An Introduction to Ergodic Theory, page 153, Remark (2), he writes
If $E(X,T)$ denotes the set of extreme points of $M(X,T)$ then for each $\mu \in M(X,T)$ there is a unique measure $\tau$ on the Borel subsets of the compact metrisable space $M(X,T)$ such that $\tau(E(X,T))=1$ and $\forall f \in C(X)$ $\int_{X} f(x) \ d\mu(x) = \int_{E(X,T)} \left( \int_{X} f(x) \ dm(x)\right) \ d\tau(m).$ We write $\mu = \int_{E(X,T)} m \ d\tau(m)$ and call this the ergodic decomposition of $\mu$.
Here $X$ is a compact metric space, $T:X\to X$ is a continuous, $M(X,T)$ is the space of $T$-invariant ($\mu\circ T^{-1} = \mu$) Borel probability measures, and $C(X)$ is the space of continuous, real-valued functions on $X$.
Why is $E(X,T)$ a Borel set? In this paper, the author points out on page 5 that the set of extremal points is not Borel in general. He then claims that he'll give sufficient conditions that guarantee set of extremal will be Borel, but doesn't seem to. Does he give such conditions? If so, where?
By saying "we write $\mu = \int_{E(X,T)} m \ d\tau(m)$", Walters seems to be introducing shorthand notation. Indeed, since the integrand $f(m)=m$ is not real- (nor complex-) valued, that integral doesn't make sense. Why is the notation $\mu = \int_{E(X,T)} m \ d\tau(m)$ appropriate? In particular, I suspect that experts have some intuition about this decomposition that fits this notation, but I cannot see what it is.