Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula: $$ \int\limits_{A} f(Dx) \; dH^{s}(x) = \int\limits_{ D A} f(y) \; D_*H^{s}(y) $$$$ \int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y) \; \mathrm{d}D_*H^{s}(y)\ , $$ Wherewhere $D_{*}H^{s}(M) = H^{s}(D^{-1}M)$ is the pushforward of the Hausdorff measure. Is it possible to find such a function $a(x)$ that, $$ \int\limits_{ D A} f(y) \; dD_{*}H^{s}(y) = \int\limits_{ D A} f(y) a(y) \; dH^{s}(y) $$$$ \int\limits_{ D A} f(y) \; \mathrm{d}D_{*}H^{s}(y) = \int\limits_{ D A} f(y) a(y) \; \mathrm{d}H^{s}(y)\ ? $$
I'm very interested in a usable general change of variables formula; does that exist?
