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  "body": "\n## Definition\n\nLet $G$ be a real [[Lie group|7JBRGCVJRB]] of dimension $n$, with identity $e$, and for $g \\in G$ let $L_g(h) = gh$.\nFix a nonzero alternating $n$-linear form $\\omega_e$ on the [[tangent space|E7A5DHFT01]] $T_eG$.\nLet $\\omega$ be the left-invariant $n$-form on $G$ given by $\\omega_g = (L_{g^{-1}})^* \\omega_e$, a nowhere-vanishing [[differential form|0E3G0GRPZ6]] of top degree.\nThe *left Haar measure* of $\\omega$ is the Borel measure $\\mu_\\omega$ on $G$ such that\n$$\\int_G f \\, d\\mu_\\omega = \\int_G f \\, \\abs{\\omega}$$\nfor every continuous function $f$ with compact support.\nHere $\\abs{\\omega}$ is the density associated with $\\omega$.\n\n## Remarks\n\nSince $L_h^* \\omega = \\omega$ for all $h \\in G$, the measure $\\mu_\\omega$ is invariant under left translations, so it is a left [[Haar measure|VY0HXD3GC4]] of $G$; changing $\\omega_e$ multiplies it by a positive constant.\n\nExample: on $G = GL_n(\\R)$, open in $M_n(\\R) = \\R^{n^2}$, the measure $\\abs{\\det X}^{-n} \\, dX$, with $dX$ Lebesgue measure, is a left Haar measure.\nLeft multiplication $X \\mapsto AX$ is linear with determinant $(\\det A)^n$, which cancels the change of $\\abs{\\det X}^{-n}$.\n\nExample and non-example: on $G = \\set{(a, b) : a \u003e 0, b \\in \\R}$ with $(a, b)(a', b') = (aa', ab' + b)$, the measure $a^{-2}\\, da\\, db$ is a left Haar measure.\nThe measure $a^{-1}\\, da\\, db$ is not left-invariant: left translation by $(c, 0)$ maps $(a, b)$ to $(ca, cb)$, with Jacobian $c^2$, and multiplies $a^{-1}\\, da\\, db$ by $c$.\nIt is a right Haar measure instead.\n\nA right Haar measure arises in the same way from right-invariant forms.\nThe group is unimodular, meaning that left and right Haar measures coincide, if and only if $\\abs{\\det \\operatorname{Ad}(g)} = 1$ for all $g \\in G$, where $\\operatorname{Ad}$ is the [[adjoint representation|P6ZP0TE07M]].\n",
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