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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.

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Maybe for measures that are absolutely continuous wrt the Lebesgue measure? –  copper.hat Jun 1 '12 at 6:01
That's a good thought. Certainly, Radon-Nikodym could be used to generalize it to additional measures. Nonetheless, I'm still curious if there's something intrinsic to the Lebesgue measure that's required for the formula to hold. –  Oyqcb Jun 1 '12 at 6:07
I suspect an intimate connection between the Lebesgue measure and the determinant. I would imagine any significant generalization (ie, beyond invoking Radon-Nikoydm) would need to concoct an appropriate equivalent of a determinant for the measure in question. –  copper.hat Jun 1 '12 at 6:10
It is the general linear group action on $\mathbb{R}^n$ and the homogenity of $\mathbb{R}^n$ which makes that case so special. You may want to have a look at the Haar measure. –  user20266 Jun 1 '12 at 6:30
You may also want to have a look at Hausdorff measure, area and coarea formula. –  user20266 Jun 1 '12 at 6:41

3 Answers 3

please have a look at the monograph by Patric Muldowney theory of Random variation John Wiley and sons. it suggests a formuala and proves using Henstock-kurzweil apparoach

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Also you can have look on V.I. Bogachev. "Measure Theory."

In the case you are interested in probability theory, see R. Durrett, "Probability: Theory and Examples", 4th ed, 2010, pp 30-31.

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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 $|\mathrm{det} DF|\mathrm{d}\lambda$ under $F$ is $\mathrm{d}\lambda$).

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