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 $\phi\in\ell^\infty(G)$?
> 
> 
> 



Amenability with respect to a function
The answer is no, no such non-amenable group can exist. It follows from Justin Moore's answer to his own [question](https://mathoverflow.net/questions/60247/when-is-non-amenablity-witnessed-by-a-single-non-measurable-set) that a single characteristic function can witness the non-amenability of a group.


