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][1], 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.

  [1]: https://arxiv.org/pdf/1705.04091.pdf