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  "body": "\n## Definition\n\nLet $G$ be a group acting on a set $X$ by a [[group action|0WCY42EV9V]] $(g, x) \\mapsto g \\cdot x$.\nLet $\\ell^\\infty(X)$ be the space of bounded functions $X \\to \\R$, and for $g \\in G$ and $f \\in \\ell^\\infty(X)$ let $(g \\cdot f)(x) = f(g^{-1} \\cdot x)$.\nA [[mean|ZYNY2VFECQ]] on $X$ is a linear map $m \\colon \\ell^\\infty(X) \\to \\R$ such that:\n\n1. $m(f) \\ge 0$ whenever $f \\ge 0$.\n2. $m(1) = 1$ for the constant function $1$.\n\nThe action is *amenable* if there is a mean $m$ on $X$ with $m(g \\cdot f) = m(f)$ for all $g \\in G$ and $f \\in \\ell^\\infty(X)$.\n\n## Remarks\n\nExample: suppose the action has a finite orbit $O = G \\cdot x_0$.\nThen $m(f) = \\frac{1}{\\abs{O}} \\sum_{x \\in O} f(x)$ is a mean, and it is invariant because each $g \\in G$ permutes $O$.\nSo every action with a finite orbit is amenable, whatever the group.\n\nNon-example: the free group $F_2$ acting on itself by left multiplication is not amenable.\nIndeed, an invariant mean for this action is exactly what makes a group [[amenable|CEAT9BESKK]], and $F_2$ is not amenable.\n\nA group $G$ is amenable if and only if its action on itself by left multiplication is amenable.\nIf $G$ 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 $X$.\n\nThe name is also used in other senses.\nIn ergodic theory, an amenable group action often just means an action of an amenable group.\nZimmer'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.\n",
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