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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]], with no topology, and let $\\mathcal{P}(G)$ be the set of all subsets of $G$.\nThe group $G$ is *amenable* if there is a function $\\mu \\colon \\mathcal{P}(G) \\to [0, 1]$ such that:\n\n1. $\\mu(G) = 1$.\n2. $\\mu(A \\cup B) = \\mu(A) + \\mu(B)$ for all disjoint $A, B \\subseteq G$.\n3. $\\mu(gA) = \\mu(A)$ for all $g \\in G$ and $A \\subseteq G$.\n\nSuch a $\\mu$ is a left-invariant finitely additive probability measure on $G$.\n\n## Remarks\n\nExample: a finite group $G$ is amenable, with $\\mu(A) = \\abs{A} / \\abs{G}$; left translation is a bijection, so $\\abs{gA} = \\abs{A}$.\n\nNon-example: the [[free group|0KTW26C80J]] $F_2$ on $a, b$ is not amenable.\nLet $W(x)$ be the set of reduced words that begin with the letter $x$.\nThen $F_2 = W(a) \\sqcup a W(a^{-1})$ and $F_2 = W(b) \\sqcup b W(b^{-1})$.\nAn invariant $\\mu$ would give $\\mu(W(a)) + \\mu(W(a^{-1})) = 1$ and $\\mu(W(b)) + \\mu(W(b^{-1})) = 1$, but these four disjoint sets lie in $F_2$, so their total is at most $1$, a contradiction.\n\nEquivalently, there is a left-invariant mean on $\\ell^\\infty(G)$, a positive linear functional $m$ with $m(1) = 1$ and $m(g \\cdot f) = m(f)$.\nFor a locally compact group, one asks instead for a left-invariant mean on $L^\\infty(G)$; for discrete groups the two notions agree.\nFor a countable group, amenability is equivalent to the Følner condition.\nSubgroups, quotients and extensions of amenable groups are amenable, and abelian and solvable groups are amenable.\n",
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