I have a reference request concerning Proposition 1.6 in the following article http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183548783

The setting: Let $G$ be a locally compact, second countable group. Let $S = (S, \mu)$ be a Polish space. Assume we have a Borel measurable action $G \times S \rightarrow S$. Assume that $\mu$ is quasi invariant.

The statement: There is a standard measure space $E = (E, \nu)$ with G invariant measurable map $\phi :S \rightarrow E$ such that $\phi_*(\mu)=\nu$ and $\mu = \int^\oplus \mu_y d \nu(y)$, where $\mu_y$ is supported on $\phi^{-1}(y)$, $\mu_y$ quasi invariant and ergodic for almost all $y$.

I have tried to recover this result via Choquet theory, but I am not sure what topology to put on the quasi invariant measures, since measure classes are not closed in the $*$ topology. What is the right topology on quasi invariant Radon measures, such that they form a locally compact convex subset of a topological vectorspace?

Additional question: If the action is topological, say $E$ is a Polish space, and smooth in the sense that $G \backslash X$ is $T_0$ or equivalently almost Hausdorff, how can we relate $E$ and $G \backslash X$? Since $E$ is Hausdorff and $G \backslash X$ only almost Hausdorff, I am not sure how to relate the topologies.