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Let $(X_1 \times X_2,d\mu)$ be a measure space with $X_2$ compact. Suppose that we have a continuous (diagonal) action of a topological group $G$ on $X=X_1 \times X_2$. I know that the action of $G$ on $X_2$ is ergodic (with respect to the ''projection'' measure on $X_2$) and I would like to show that the action of $G$ on $X$ is ergodic. Are there any references or tools that might be useful here ? Any help or reference would be appreciated.

PS: We also have an action of a discreet subgroup $\Gamma \subset G$ on $X$ and $\Gamma \backslash X_1$ admits a Baily-Borel compactification. I had the idea to use Stone–Weierstrass theorem but it does not seem to work.

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    $\begingroup$ If e.g. $X_1$ and $X_2$ are copies of $G:=\mathbb R/Z$ with the standard structure, isn't any set $\{(x,y):x-y\in A\}$ invariant, for any fixed $A\subset\mathbb G$? What am I missing? $\endgroup$
    – Pietro Majer
    Commented Sep 5, 2021 at 7:41
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    $\begingroup$ You mention Baily-Borel compactification, so I suppose your group $G$ is more then just a locally compact group. Is it a semisimple Lie group? Are you aware of Howe-Moore theorem? $\endgroup$
    – Uri Bader
    Commented Sep 6, 2021 at 12:48
  • $\begingroup$ Thank you, I will look it up and see if if it's helpful. The Group $G$ is subgoup of a a semisimple Lie group. $\endgroup$
    – Osheaga
    Commented Sep 15, 2021 at 21:19
  • $\begingroup$ If $G$ is contained in a semisimple Lie group $H$ and the action on one of the spaces, say $X_1$ extends to $H$ then the $G$-action on $X$ will be ergodic. This is because the $H$-action on $X_1$ will be mixing, hence also the $G$-action. $\endgroup$
    – Uri Bader
    Commented Sep 16, 2021 at 8:31

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