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  "body": "\n## Definition\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]] and let $\\mathcal{B}$ be the $\\sigma$-algebra of [[Borel sets|0WQXY32Q8D]] of $G$.\nA *left Haar measure* on $G$ is a measure $\\mu$ on $\\mathcal{B}$ such that:\n\n1. $\\mu$ is a [[Radon measure|HZNEQQBCW4]]: it is finite on compact sets, outer regular on Borel sets and inner regular on open sets;\n2. $\\mu$ is not the zero measure;\n3. $\\mu$ is left-invariant: $\\mu(gA) = \\mu(A)$ for every $g \\in G$ and every $A \\in \\mathcal{B}$, where $gA = \\set{ga : a \\in A}$.\n\nA *right Haar measure* is defined in the same way with $\\mu(Ag) = \\mu(A)$ in clause 3.\n\n## Remarks\n\nExample: Lebesgue measure $\\lambda$ on $(\\R^n, +)$.\nIt is finite on compact sets, regular and nonzero, and $\\lambda(x + A) = \\lambda(A)$ for every $x \\in \\R^n$ and Borel set $A$.\n\nNon-example: the Dirac measure $\\delta_0$ on $(\\R, +)$ is a nonzero Radon measure but fails clause 3, since $\\delta_0(\\set{0}) = 1$ and $\\delta_0(1 + \\set{0}) = \\delta_0(\\set{1}) = 0$.\n\nIn integral form, left invariance says $\\int_G f(gx) \\, d\\mu(x) = \\int_G f(x) \\, d\\mu(x)$ for every $g \\in G$ and every compactly supported continuous $f$.\nIf $\\mu$ is a left Haar measure, then $A \\mapsto \\mu(A^{-1})$ is a right Haar measure.\nA left Haar measure gives positive measure to every nonempty open set.\n",
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