# Amenability with respect to a function

Let $(G,\cdot)$ be a group and $\phi:G\rightarrow\mathbb R$ bounded. Let me say that the pair $(G,\phi)$ is amenable if there is a finitely additive probability measure $\mu$ on $G$ such that for all $y\in G$

$$\int \phi(x)d\mu(x)=\int \phi(x\cdot y)d\mu(x)=\int\phi(y\cdot x)d\mu(x)$$

Question: Does there exist a non-amenable group such that the pair $(G,\phi)$ is amenable for all $\phi\in\ell^\infty(G)$?

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How about the free group with 2 generators? –  Mark Sapir Sep 4 '11 at 15:05
I have no idea. Is it trivial that for any $\phi$ there is such a measure? –  Valerio Capraro Sep 4 '11 at 15:28
Did you try any $\phi$ at all? –  Mark Sapir Sep 4 '11 at 15:57
There are ergodic theorems for free group actions (Bufetov, Alexander I. Convergence of spherical averages for actions of free groups. Ann. of Math. (2) 155 (2002), no. 3, 929–944.) Perhaps that can help. But I would suggest that you ask the simplest question for which you do not know the answer first. How about the indicator function of a generator? –  Mark Sapir Sep 4 '11 at 17:49
I think you should try to consider the indicator case yourself. It should be an easy exercise. Then you will get an idea what to do next. –  Mark Sapir Sep 4 '11 at 23:32