Is there a change of variables formula for integration using a generic measure?
Is there a change of variables formula for a measure theoretic integral that uses a generic measure? Specifically, a typical change of variable of variables formula has the form
where , denotes the n-dimensional Legbesgue measure, and denotes the Jacobian of . Basically, this formula is locked to the Lebesgue measure and I'd like to know if it's possible to replace that measure with a generic measure. Certainly, we could use Radon-Nikodym to generalize it to a few extra measures. I also believe that Federer has generalized this to something similar with the Hausdorff measure. Mostly, I'd like to know if there's something special about the Lebesgue measure that restricts the change of variables formula to this measure. On a similar note, is there an infinite dimensional analogue?