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.