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Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space. Assume that $G$ acts on $X$ by homeomorphisms. Consider the following two definitions:

  1. [$C^*$-algebras and finite dimensional approximations- Brown and Ozawa- Definition 4.3.5.]

The action is (topologically) amenable if there exists a net of continuous maps $m_i: X\to Prob(G)$ s.t. for each $s\in G$: $$ \lim_{i\to \infty} (\sup_{x\in X} \|s.m_i^x-m_i^{s.x}\|_1)=0$$ where $s.m_i^x(g)=m_i^x(s^{-1}g)$.

Now, regard $X$ as a $G$-set. Then one can define:

2.[Amenability of Groups and G-Sets, see definition in the introduction, for example]:

The action is amenable if there exists an invariant mean on the power set of $X$.

I would expect that (1) will imply (2). Namely, that if the action of $G$ on $X$ is topologically amenable then it is amenable (set-theoretically).

However, I know very little about amenable actions and I would appreciate any help.

Thanks.

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    $\begingroup$ I usually call 1 "Zimmer-amenable" and 2 "amenable". Actually the suffix -able is not well-suited to definition 1, and I think of def 1 as "ameaning" (moyennant in French, vs moyennable in sense 2). Introducing a confusing definition is somewhat hopelessly irreversible. Fortunately the definitions are so opposite that the confusion is somewhat limited, in the sense that any interesting statement using one definition is trivially false or tautologic in the other direction. $\endgroup$ – YCor Dec 9 '18 at 20:55
  • $\begingroup$ @YCor Did anyone really use "moyennant" in this context? $\endgroup$ – R W Dec 9 '18 at 21:34
  • $\begingroup$ @RW I don't think so. But I would if I had to use this property at some point. $\endgroup$ – YCor Dec 9 '18 at 21:46
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The terminology is unfortunately confusing as these properties are in a sense "orthogonal". The definition in terms of existence of an invariant mean on the action space (your definition 2) goes back to von Neumann, whereas your definition 1 goes back to Zimmer who introduced it (in the measure category and in a somewhat different form though) in 1977. Amenability of the group $G$ is equivalent to the amenability in the sense of Definition 1 of its trivial action on the one-point space and to the amenability in the sense of Definition 2 of its action on itself.

For an explicit counterexample to your claim take the boundary action of a free group. It is topologically amenable in the sense of Definition 1, but not amenable in the sense of Definition 2.

A counterexample in the opposite direction is provided just by the trivial action of any non-amenable group on a singleton.

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