I have a feeling the Disintegration Theorem and the Co-Area formula for Lipschitz functions are actually very much related but I cannot seem to formalize it.

DISINTEGRATION THEOREM: Let $(\mathsf{X}, \mathcal{X}, \mu)$ be a probability space and $(\mathsf{Y}, \mathcal{Y})$ be a measurable space. Let $\xi:\mathsf{X}\to\mathsf{Y}$ be a random variable with distribution $\nu = \mu \circ \xi^{-1}$. Then $\nu$-almost everywhere there exists a uniquely determined family of probability measures $\{\mu_y\}_{y\in\mathsf{Y}}$ on $(\mathsf{X}, \mathcal{X})$ such that (...skipping some results...) for any Borel measurable function $f:\mathsf{X}\to[0, \infty]$ the following is satisfied: $$ \int_{\mathsf{X}} f(x) d\mu(x) = \int_{\mathsf{Y}}\int_{\xi^{-1}(y)} f(x) d\mu_y(x) d\nu(y) $$

CO-AREA FORMULA: Let $\xi:\mathbb{R}^m\to\mathbb{R}^n$ be Lipschitz with $m > n$ and $f:\mathbb{R}^m\to\mathbb{R}$ be Lebesgue measurable with generalized Jacobian $J_n\xi(x) = \left|D\xi(x) D\xi(x)^\top\right|^{1/2}$. Then $$ \int_{\mathbb{R}^m} f(x) \lambda^m(dx) = \int_{\mathbb{R}^n}\int_{\xi^{-1}(y)} f(x) J_n\xi(x)^{-1}\mathcal{H}^{m-n}(dx) \lambda^n(dy) $$ where $\lambda^n$ is the $n$-dimensional Lebesgue measure and $\mathcal{H}^{n-m}$ is the $(n-m)$-dimensional Hausdorff measure on $\xi^{-1}(y)$.

Attempted Solution

The disintegration theorem seems to provide sufficiency conditions for this type of decomposition to happen, whereas the Co-Area formula gives an expression for the measures $\mu_y$ in a specific case. My guess is that one should take:

• $(\mathsf{X}, \mathcal{X}) = (\mathbb{R}^m, \mathcal{B}(\mathbb{R}^m))$
• $(\mathsf{Y}, \mathcal{Y}) = (\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$
• $\mu = \lambda^m$
• $\nu = \xi_*\lambda^m$

The only way in which I can conciliate the two formulas is if the following is somehow true $$ d\mu_y(x)d\nu(y) = J_n\xi(x)^{-1}d\mathcal{H}^{m-n}(dx)\lambda^n(dy). $$ Not sure how to show it. Somehow I have a feeling that the Radon-Nikodym derivative of the pushed-forward Lebesgue measure and the Lebesgue measure gives a Jacobian $$ \frac{d \xi_*\lambda^m}{d\lambda_m} = J_n \xi(x). $$ When $\xi$ is a diffeomorphism this would be coherent with the change of variables formula found in multivariable calculus where $$ \frac{d \xi_*\lambda^m}{d\lambda_m} = |D\xi(x)| $$