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:

- [$C^*$-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.

ameaning" (moyennantin French, vsmoyennablein 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