Topological amenability of actions - forgetting topology

Question

Let $G$ be a (countable) discrete group and let $X$ be a locally compact Hausdorff space. Assume that $G$ acts on $X$ by homeomorphisms. Recall that the action is (topologically) amenable if there exists a net of continuous maps $m_i: X\to \mathrm{Prob}(G)$ s.t. for each $s\in G$ and $K\subseteq X$ compact: $$\lim_{i\to \infty} (\sup_{x\in K} \|s\cdot m_i^x-m_i^{s.x}\|_1)=0$$ where $s\cdot m_i^x(g)=m_i^x(s^{-1}g)$, and $\mathrm{Prob}(G)$ denotes the space of (positive) probability measures on $G$. See Definition 2.1 in https://www.ams.org/journals/tran/2002-354-10/S0002-9947-02-02978-1/S0002-9947-02-02978-1.pdf

It is clear that if the $G$-action on $X$ is (topologically) amenable, then it is also amenable when we forget the topology of $X$, that is, replace its topology by the discrete topology.

I would like to known if the converse is also true: if the $G$-action on $X_d$ (meaning $X$ with the discrete topology) is amenable, is the original $G$-action on $X$ (with its own topology) also amenable?

Answer 1

After posting the question, I discussed with Siegfried Echterhoff, and more or less at the same time Nicolas Monod sent me an email, both giving me the following argument:

for an action of a group $G$ on a discrete space $X$, amenability of the action is equivalent to the statement that all stabilizer subgroups $$G(x):=\{g\in G: gx=x\}$$ are amenable.

So, for instance, every free action is amenable in this sense. But there are many examples of free actions that are not topologically amenable. For instance, take any non-amenable group $G$ (e.g. $G$ the free group $F_2$) that embeds (as a subgroup) into a compact group $K$ (e.g. $K=O(3)$), and consider the translation $G$-action on $K$. This is a free action, so amenable for $K$ with the discrete topology, but not topologically amenable because $K$ is compact and therefore it carries an invariant measure, so it follows that the $G$-action cannot be topologically amenable, as topologically amenable actions do not carry invariant measures, unless the acting group is amenable.