Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as well? That is:


***Is the residual amenability of a discrete finitely generated group $G$ equivalent to the existence of a function $\mu \colon \mathcal{P}(G) \to \mathbb{R}$ satisfying certain properties?***



Characterizing residually amenable groups
It is easy to see that a finitely generated group is residually amenable if and only if there exists an bi-invariant ultra-metric on $G$ and a finitely additive $G$-invariant measure on open (with respect to the metric) subsets of $G$.


