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  "body": "\n## Definition\n\nLet $(X, \\Sigma)$ be a measurable space and let $\\mu$ be a [[measure|PR11BZQQPG]] on $\\Sigma$.\nLet $G$ be a group that acts on $X$ by [[group action|0WCY42EV9V]], such that each map $T_g \\colon X \\to X$, $x \\mapsto g \\cdot x$, is measurable.\nThe measure $\\mu$ is *quasi-invariant* under $G$ if for every $g \\in G$ and every $A \\in \\Sigma$,\n$$\\mu(A) = 0 \\iff \\mu(T_g^{-1}(A)) = 0.$$\n\n## Remarks\n\nEquivalently, for each $g$ the [[pushforward|3N66TM3TSQ]] $(T_g)_* \\mu$ and $\\mu$ are each [[absolutely continuous|P0P21KDV6E]] with respect to the other.\nEvery invariant measure is quasi-invariant.\n\nExample: let $G = \\R$ act on $X = \\R$ by translations, and let $\\mu$ be the standard [[Gaussian measure|P0DV2TCH1N]].\nIts density is positive everywhere, so $\\mu(A) = 0$ exactly when $A$ has Lebesgue measure $0$.\nLebesgue [[null sets|VQQM4Y550T]] are preserved by translations, so $\\mu$ is quasi-invariant.\n\nNon-example: under the same action, the [[Dirac measure|0ACRGYNHSR]] $\\delta_0$ is not quasi-invariant.\nFor $g = 1$ and $A = \\set{1}$, one has $\\delta_0(A) = 0$ but $\\delta_0(T_1^{-1}(A)) = \\delta_0(\\set{0}) = 1$.\n\nBy the [[Cameron–Martin theorem|WAE4ZGJXJ7]], Wiener measure is quasi-invariant under translations by the Cameron–Martin space, but not under all translations of the path space.\n\n## Explanations\n\n### L1 human\n\nMoving a set by the group never turns a negligible set into a non-negligible one.\nThe measure can change in size, but its null sets stay the same.\n",
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