Let $G$ be a discrete nonamenable countable group acting on a standard probability space $(X,\mu)$ through measure-preserving transformations.


Is the action of $G$ always amenable?


(Amenable action, in the sense of Zimmer, which is equivalent to the action being orbit-equivalent to some action of the integers on $(X,\mu)$)


I do not see a proof, and am unfortunately unable to find a counterexample either. The closest thing I get, though perhaps not quite related, is that [Bowen and Nevo](https://doi.org/10.48550/arXiv.1103.2519) proved the existence of pointwise ergodic families for amenable equivalence relations (and hence amenable actions), and [Tao](https://doi.org/10.48550/arXiv.1505.04725) has an example of an action of $\mathbb{F}\_2$ where the family of increasing subsets of words of the same length does not form a pointwise ergodic family.



Nonamenable p.m.p. action on a standard probability space
A measure preserving action of a non-amenable group on a finite measure space is always **non-amenable**, if this is what you meant.


It is pretty obvious if one uses the most natural definition of an amenable action in terms of asymptotically equivariant families of probability measures. This definition (due to Renault) is equivalent to the orginal definition of Zimmer (given in terms of a fixed point property), but is much more constructive and easier to deal with (this is the reason why some of the first papers on amenability of actions are much longer than necessary). The definition you use is a combination of either of these definitions with a number of quite deep results culminating in the Connes - Feldman - Weiss theorem.


Indeed, according to Renault definition, an action of a countable group $G$ on a measure space $(X,m)$ is amenable if there exists a family of measurable maps $x\mapsto\lambda\_n^x$ from $X$ to the space of probability measures on $G$ that is **asymptotically equivariant** in the sense that
$\|g\lambda\_n^x - \lambda\_n^{gx}\|\to 0$ almost surely for any $g\in G$. If the measure $m$ is invariant, then one can easily verify that the averages $\lambda\_n=\int\lambda\_n^x\,dm(x)$ are asymptotically invariant (i.e., $\|g\lambda\_n-\lambda\_n\|\to 0$ for any $g\in G$), and therefore the group $G$ is amenable.


By using Renault's definition one also immediately sees that **any** (no matter measure preserving, or not) action of an amenable group is amenable (just take an asymptotically invariant sequence $\lambda\_n$ on $G$ and put $\lambda\_n^x\equiv\lambda\_n$). Thus, for probability measure preserving actions amenability of the acting group is equivalent to amenability of the action. However, for actions without an invariant measure the situation is different, and there exist amenable actions of non-amenable groups.


PS Ergodic theorems *per se* are not that much related to amenability of actions. What is more important is isoperimetric properties of the action, which, indeed, can be used to characterize amenability (in a way similar to the isoperimetric characterization of amenable groups).


