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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?2012-06-01
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    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.2012-06-01
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    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.2012-06-01
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    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.2012-06-01
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    You may also want to have a look at Hausdorff measure, area and coarea formula.2012-06-01
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    The change-of-variables formula for abstract measure spaces can be considered to be the Radon-Nikodym theorem. http://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem2014-01-14

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