Let $G$ be a finitely generated group. Does the following condition imply the amenability of $G$: there is a function $\mu:\mathcal{P}(G) \to [0,1]$ such that:
A group distinguishing this condition from amenability cannot contain $F_2$. I am aware of the relationship to the (now solved) Maharam problem.
I am expecting an answer so much as asking whether (and where) this question has been studied.