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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $\\ell^\\infty(G)$ be the real vector space of bounded functions $f \\colon G \\to \\R$.\nFor $g \\in G$ and $f \\in \\ell^\\infty(G)$, let $g \\cdot f$ be the function $x \\mapsto f(g^{-1} x)$.\nA *left-invariant mean* on $G$ is a [[linear form|07YQ1C78G0]] $m \\colon \\ell^\\infty(G) \\to \\R$ with these three properties.\n\n1. Positivity: $m(f) \\ge 0$ for every $f \\ge 0$.\n2. Normalization: $m(\\mathbf{1}) = 1$, where $\\mathbf{1}$ is the constant function $1$.\n3. Invariance: $m(g \\cdot f) = m(f)$ for every $g \\in G$ and $f \\in \\ell^\\infty(G)$.\n\n## Remarks\n\nRight-invariant and two-sided invariant means are defined in the same way.\nA group has a left-invariant mean if and only if it is [[amenable|CEAT9BESKK]].\nA left-invariant mean $m$ gives a left-invariant [[finitely additive measure|KFWQ24D9M6]] $\\nu(A) = m(\\mathbf{1}_A)$ on all subsets of $G$, with $\\nu(G) = 1$, and every such $\\nu$ arises in this way.\n\nExample: for a finite group $G$, the average $m(f) = \\frac{1}{\\abs{G}} \\sum_{x \\in G} f(x)$ is a left-invariant mean.\nPositivity and normalization are clear, and invariance holds because $x \\mapsto g^{-1} x$ is a bijection of $G$.\n\nNon-example: on $G = \\Z$, the evaluation $m(f) = f(0)$ is positive and normalized but not invariant.\nFor $f = \\mathbf{1}_{\\set{0}}$ and $g = 1$, the function $g \\cdot f$ is $\\mathbf{1}_{\\set{1}}$, so $m(g \\cdot f) = 0 \\ne 1 = m(f)$.\n\nOn $\\Z$, invariant means exist by the Hahn–Banach theorem, for example Banach limits, but none can be given by an explicit formula.\nFor a locally compact group, one uses $L^\\infty(G)$ with respect to Haar measure in place of $\\ell^\\infty(G)$.\n\n## Explanations\n\n### L1 human\n\nAn invariant mean is an averaging rule for all bounded functions on a group that ignores translations.\nIt always exists on finite groups and abelian groups, and never on free groups of rank two.\n",
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