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  "body": "\n## Statement\n\nLet $G$ be a locally compact Hausdorff topological group.\n\n1. There is a nonzero [[Radon measure|HZNEQQBCW4]] $\\mu$ on the Borel sets of $G$ that is left-invariant: $\\mu(gA) = \\mu(A)$ for every $g \\in G$ and every Borel set $A \\subseteq G$.\n2. If $\\nu$ is another nonzero left-invariant Radon measure on $G$, then $\\nu = c \\mu$ for some constant $c \u003e 0$.\n\n## Remarks\n\nSuch a measure is a left Haar measure of $G$.\nOne proof compares the size of a compact set $K$ with the number of translates of a small open set needed to cover it.\nThese ratios have a limit point, by a compactness argument, which gives an invariant content and then a Radon measure.\nAnother proof builds an invariant positive linear functional on compactly supported continuous functions and applies the [[Riesz–Markov–Kakutani representation theorem|DWFPDF8XTB]].\nUniqueness follows from a Fubini argument applied to $\\mu \\otimes \\nu$ and the map $(x, y) \\mapsto (x, xy)$.\n\nExample: on $G = (\\R^n, +)$, Lebesgue measure is a Haar measure, so every translation-invariant Radon measure on $\\R^n$ is a multiple of Lebesgue measure.\nOn the circle group, the Haar measures are the multiples of arc length.\nOn a discrete group, the Haar measures are the multiples of counting measure.\n\nLocal compactness enters the proof through compact sets with nonempty interior, which give the comparison of sizes.\n\n## Explanations\n\n### L1 human\n\nEvery locally compact group has a natural notion of volume that does not change when sets are moved by the group.\nIt is unique up to the choice of a unit.\nFor the real line it is ordinary length, and for a finite group it is counting.\n\n### L2 human\n\nHaar measure lets one integrate over a group, which is the base of harmonic analysis on groups.\nIt gives the group algebra $L^1(G)$, the Peter–Weyl theorem for compact groups and Pontryagin duality for abelian groups.\nFor a compact group the measure is finite and is usually normalized to total mass $1$.\nLeft and right Haar measures may differ, and their ratio is the modular function of $G$.\n",
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