Let $G$ be an infinite countable group. If $G$ is amenable then it has a number of other interesting properties. To prove such a property from the existence of an invariant mean on $G$, we usually start with any invariant mean and use it for some kind of averaging, or exploit its invariance.
Question: Are there any proofs of properties of $G$ in which we cannot use an arbitrary invariant mean? So the proof works with some invariant means on $G$ but not with others?