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High Quality Content by WIKIPEDIA articles! In mathematics, a quasi-invariant measure ? with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function by T. An important class of examples occurs when X is a smooth manifold M, T is a diffeomorphism of M, and ? is any measure that locally is a measure with base the Lebesgue measure on Euclidean space. Then the effect of T on ? is locally expressible as multiplication by the Jacobian determinant of the derivative (pushforward) of T.