I'm reading an old paper by Varadarajan (1963) titled "Groups of automorphisms of Borel spaces" and I'm trying to fulfill the details. There are many questions I thought about for quite a long time with no success. For the moment, I confine myself to the first part of the proof of Theorem 4.2 (possibly, the remaining one would be treated in another question).
Notation and Terminology: Suppose $G$ lcsc (=locally compact second countable) group. By measure we mean a Borel probability measure. A measure on a Borel space $X$ is said to be invariant if $\mu(A)=\mu(gA)$ for all borel subsets $A$ and $g\in G$ and we write $\mu\in\mathscr{I}$. It is ergodic if all $G$-invariant Borel subsets are either null or conull. The set of ergodic measures is denoted by $\mathscr{E}$.
Q: Suppose $Y$ is a compact metric $G$-space ($G$ acts continuously on $Y$), so that the space $\mathscr{I}_Y$ of $G$-invariant measures on $Y$ is compact wrt the weak-topology (the one which makes continuous the functions $\mu\mapsto \int_Y g\,d\mu$ forall bounded continuous real-valyued functions $g\in C_b(Y)$). Then, referring to a paper by Choquet titled "Existence et unicité des représentations intégrales", he claims $\mathscr{E}_Y$ is a $G_\delta$ subset. Unfortunately, I have not found such a paper, so I would like to know how to prove that $\mathscr{E}_Y$ is $G_\delta$ (any useful reference on the subject is appreciated, even better if a book).
Edit: I have made the notion of ergodicity more explicit.