Is there a generic change of variables formula for a measure theoretic integral that does not use the Lebesgue measure? Specifically, most references that I can find give a change of variables formula of the form:
$$
\int_{\phi(\Omega)} f d\lambda^m = \int_{\Omega} f \circ \phi |\det J_\phi| d\lambda^m
$$
where $\Omega\subset\Re^m$, $\lambda^m$ denotes the $m$-dimensional Lebesgue measure, and $J_\phi$ denotes the Jacobian of $\phi$. Is it possible to replace $\lambda^m$ with a generic measure and, if so, is there a good reference for the proof? I'm also curious if a similar formula holds in infinite dimensions.
Best Answer
Given a measure space $(X_1,M_1,\mu)$ and a measureable space $(X_2,M_2)$ you can define the pushforward measure on $M_2$ of $\mu$ by a measurable function $F:X_1\to X_2$ to be $F\mu(E)=\mu(F^{-1}(E))$. Then you have the formula
$$\int_{X_2}g\;\mathrm{d}F\mu=\int_{X_1}g\circ F\;\mathrm{d}\mu$$
which is effectively the change of variables between the measure spaces $(X_1,M_1,\mu)$ and $(X_2,M_2,F\mu)$. The change of variables with Lebesgue measure should then a special case of this (the pushforward of $|\mathrm{det} DF|\lambda$ under $F$ is $\lambda$).