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  "body": "\n## Definition\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]] and let $\\mu$ be a left [[Haar measure|VY0HXD3GC4]] on $G$.\nFor $g \\in G$, the measure $\\mu_g(A) = \\mu(Ag)$ on Borel sets $A$ is again a left Haar measure.\nBy the uniqueness part of the [[existence and uniqueness of Haar measure|KB792QTG4F]], there is a unique real number $\\Delta(g) \u003e 0$ such that\n$$\\mu(Ag) = \\Delta(g) \\, \\mu(A) \\quad \\text{for every Borel set } A \\subseteq G.$$\nThe map $\\Delta \\colon G \\to (0, \\infty)$ is the *modular function* of $G$.\n\n## Remarks\n\nThe number $\\Delta(g)$ does not depend on the choice of $\\mu$, since any other left Haar measure is $c\\mu$ with $c \u003e 0$.\nThe map $\\Delta$ is a continuous homomorphism into the multiplicative group $(0, \\infty)$.\nIn integral form, $\\int_G f(xg^{-1}) \\, d\\mu(x) = \\Delta(g) \\int_G f(x) \\, d\\mu(x)$ for compactly supported continuous $f$.\n\nExample: let $G$ be the group of pairs $(a, b)$ with $a \u003e 0$ and $b \\in \\R$, with product $(a, b)(c, d) = (ac, ad + b)$, the group of maps $x \\mapsto ax + b$.\nThe measure $d\\mu = da \\, db / a^2$ is a left Haar measure.\nRight multiplication by $(c, d)$ sends $(a, b)$ to $(ac, ad + b)$, a map with Jacobian determinant $c$.\nSo $\\mu(A(c, d)) = \\int_A c \\, (ac)^{-2} a^2 \\, d\\mu = c^{-1} \\mu(A)$, and $\\Delta(c, d) = 1/c$.\n\nFor this group the measure $da \\, db / a$ is a right Haar measure but not a left one, so the left and right Haar measures differ.\n\nSome authors use the reciprocal convention, defining the modular function by $\\mu(Ag) = \\Delta(g)^{-1} \\mu(A)$ or through right Haar measure, so formulas must be checked against the source.\n",
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