Let $G$ be a finitely generated (in my case also amenable) group and $f:G\to[0,1]$. Suppose that there is a finitely additive probability measure $\mu$ on $G\times G$ and a real number $L$ such that $\int f(xsy)d\mu(x,y)\geq L$, for all $s\in G$.
Question: Does there exist a finitely additive probability measure on $G$, say $\nu$, such that $\int f(xs)d\nu(x)\geq L$, for all $s\in G$?
If the group is abelian, the answer is positive: approximate $\mu$ in the weak* topology with countably additive measures $\mu_\alpha$; define $\nu_\alpha(x)=\sum_y\mu_\alpha(xy^{-1},y)$ and take $\nu$ to be a weak* limit point of $\nu_\alpha$. It works basically because I can put the $s$ after the $y$, by commutativity.
One more datum that may help is that I know something about $\mu$. Precisely, it is an iterated integral w.r.t two f.a. measures on $G$, $\sigma$ and $\lambda$, where $\lambda$ is left-invariant (this explains why in my hypotheses, $\mu$ is only centrally invariant and actually that integral is equal to $L$, for all $s$). Adding this datum, one easily sees that my question has positive answer for all amenable groups such that every left-invariant mean is also right-invariant, since the argument above still works. By an old theorem of Paterson, this class is exactly the class of groups such that every conjugacy class is finite. Therefore, my question has positive answer for all these groups and in particular for all finite groups. This is also why is not that easy to try to find a counterexample.
Thanks in advance for any help,
Valerio
