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