I am working on the following exercise from Milnor's Dynamical Systems lectures (page 3-22).
Problem 3-g: If $\mu$ is a measure on the line or circle given by $\mu(S) = \int_Sw(x)dx$ with $w(x)\geq 0,$ and if $f$ is a $C^1$ smooth map with derivative vanishing at most on a measure zero set, show that $$f_*\mu(S) = \int_S\hat{w}(y)dy\text{ with }\hat{w}(y) = \sum_{f(x) = y}w(x)/|f'(x)|$$ summed over all $x$ with $f(x) = y.$ Here $f_*\mu(S):=\mu(f^{-1}(S))$ is the standard pushforward measure.
My attempt: If we can partition $S$ into subsets on which $f^{-1}$ is one-to-one, then we may apply a change of variables. We may write: $$\mu(f^{-1}(S)) = \mu(\sum_kf^{-1}(S)\cap I_k) = \sum_k\mu(f^{-1}(S)\cap I_k)$$ But $$\mu(f^{-1}(S)\cap I_k) = \int_{f^{-1}(S)\cap I_k}w(x)dx = \int_{S}\frac{w(x)}{|f'(x)|}\chi_{I_k}dx$$ with $\chi_{I_k}$ the characteristic function on ${I_k}$. Therefore, $$\sum_k\mu(f^{-1}(S)\cap I_k) = \sum_k\int_{S}\frac{w(x)}{|f'(x)|}\chi_{I_k}dx = \int_{S}\sum_k\frac{w(x)}{|f'(x)|}\chi_{I_k} dx = \int_{S}\sum_k\frac{w(x)}{|f'(x)|}dx$$
If the basic approach is correct, then I would like to simplify and make this proof rigorous. In that direction:
- How can I ensure such a partition exists? Must it be finite?
- At what point do I apply nonnegativity of $w$?
- Are there any other gaps or issues in this proof?
- Is there an alternative approach via Radon-Nikodym?
Any help or references will be appreciated.
I think the idea of the proof is correct. You applied nonnegativity of $w$ when you used the change of variables theorem for integration, since the theorem only works for nonnegative functions and $L^1$ functions. In fact, the change of variable theorem for many to one functions almost gives the result you want. See the "A Change of Variable Theorem for Many-to-one Maps" appendix in "Measure Theory and Integration" by Michael Taylor. See the book "EG" that he references for a more general change of variables formula the works for Lipschitz maps.