I have a reference request concerning Proposition 1.6 in the following article [Link](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-6/issue-3/Ergodic-theory-group-representations-and-rigidity/bams/1183548783.full)


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.



Ergodic decomposition of quasi-invariant measure
This is a bit too long for a comment, hence I post it as an answer.


I honestly don't know where you can find a group theoretic version of ergodic decomposition proved via Choquet theory (and I'm not convinced that it exists in the setting you're interested in).


However, the exact result you quote from Zimmer is proved as Theorem 1.1 in the carefully written paper


G. Greschonig, K. Schmidt, *[Ergodic Decomposition of quasi-invariant probability measures](http://matwbn.icm.edu.pl/ksiazki/cm/cm84/cm84217.pdf)*, [Colloq. Math.](http://journals.impan.gov.pl/cm/) **84/85** (2000), part 2, 495–514, [MR1784210](http://www.ams.org/mathscinet-getitem?mr=1784210).


You'll find many interesting references in there.


For a lot of extremely helpful results that are used in and around Zimmer's work, I recommend Section 2 of


David Fisher, Dave Witte Morris, and Kevin Whyte, *[Nonergodic actions, cocycles and superrigidity](http://nyjm.albany.edu/j/2004/10-15.html)*, New York Journal of Mathematics, Volume **10** (2004) 249–269, [MR2114789](http://www.ams.org/mathscinet-getitem?mr=2114789).


I can't say anything on your final question.


