Amenable group action

Remarks

Example: suppose the action has a finite orbit $O = G \cdot x_0$. Then $m(f) = \frac{1}{\abs{O}} \sum_{x \in O} f(x)$ is a mean, and it is invariant because each $g \in G$ permutes $O$. So every action with a finite orbit is amenable, whatever the group.

Non-example: the free group $F_2$ acting on itself by left multiplication is not amenable. Indeed, an invariant mean for this action is exactly what makes a group amenable, and $F_2$ is not amenable.

A group $G$ is amenable if and only if its action on itself by left multiplication is amenable. If $G$ is amenable, every action of $G$ is amenable: for a fixed $x_0 \in X$, an invariant mean $m$ on $G$ gives the invariant mean $f \mapsto m(g \mapsto f(g \cdot x_0))$ on $X$.

The name is also used in other senses. In ergodic theory, an amenable group action often just means an action of an amenable group. Zimmer's amenability of a measure-class-preserving action is a different notion: for example, the action of $F_2$ on its boundary is amenable in Zimmer's sense.