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Let $H^{m}$ be the $m$-dimensional Hausdorff measure, let $D = \operatorname{diag }(d_1,...,d_k)$ be a nonsingular scale matrix. Consider the change of measure formula: $$ \int\limits_{A} f(Dx) \; dH^{m}(x) = \int\limits_{ D A} f(y) \; dD_{*}H^{m}(y) $$ where $D_{*}H^{m}(M) = H^{m}(D^{-1}M)$ is the pushforward of the Hausdorff measure. Is it possible to find such a function $a(x)$ that $$ \int\limits_{ D A} f(y) \; dD_{*}H^{m}(y) = \int\limits_{ D A} f(y) a(y) \; dH^{m}(y) $$

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What do you get when $k=2, m=1$? – GEdgar Apr 6 '12 at 20:44

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