Summary
12.1 In this section we define derivatives Dμ, Dμ, and D̄μ of a measure μ with respect to Lebesgue measure in V. Next, we analyze the relationships between the Lebesgue decomposition of μ and the properties of its derivatives; for example, if D̄μ is finite everywhere, μ is absolutely continuous with respect to Lebesgue measure.
12.2 First, we study the image of λ under a linear automorphism. The modulus of an automorphism of Rk is the absolute value of its determinant (Theorem 12.2.1).
12.3 In this section we prove the change of variables formula: if T is a differentiable homeomorphism from V onto W, then λ W is the image of |J(T)|λ V under T where J(T) is the Jacobian of T (Theorem 12.3.1).
12.4 This section is devoted to polar coordinates in Rn.
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© 1996 Springer-Verlag New York, Inc.
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Simonnet, M. (1996). Change of Variables. In: Measures and Probabilities. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4012-9_12
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DOI: https://doi.org/10.1007/978-1-4612-4012-9_12
Publisher Name: Springer, New York, NY
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