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:


<li>  (subadditive) $\mu(G) = 1$, $\mu(A \cup B) \leq \mu(A) + \mu(B)$, 

<li> (invariant) $\mu(A \cdot g) = \mu(A)$ for all $g$ in $G$,

<li> (exhaustive) if $\{A_n : n <\infty\}$ is pairwise disjoint, then $\inf_n \mu(A_n) = 0$.

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 not expecting an answer so much as asking whether (and where) this question has been studied.