Let $G$ be a countable discrete (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.

Thanks in advance for any help,

Valerio