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Let $G$ be a countable group with a measure preserving action on a probability measure space $(X,\mathcal{B},\mu)$. I am looking for sufficient conditions (as general as possible) so that there is a Cantor space $(C,\mathcal{C},\nu)$ (where $\nu$ is the completion of a Borel measure in $\mathcal{C}$) with a measure preserving action of $G$ on $(C,\mathcal{C},\nu)$ by homeomorphisms, and a measure preserving map $F:C\to X$ which commutes with the actions of $G$.

I get the impression that this should be true whenever $X$ is a Lebesgue-Rohlin space (the idea would be using a countable separating subset of $\mathcal{B}$ and its images by the action to define a subbasis of clopens for the topology of $\mathcal{C}$), although I have not tried to write it in detail. In any case, if there is a reference containing results like this one then it would be better to just cite it.

Motivation: I am trying to prove something for general (or as general as possible) measure preserving systems but my current techniques use weak convergence of measures so I need the action to be by homeomorphisms in a compact metric space.

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