# Reference request for a result regarding density of induced probability measure under a submersion

Let $\pi: M \to N$ be a smooth submersion from a bounded open subset of $\mathbb{R}^m$ onto $N \subset \mathbb{R}^n$, $m \geq n$. Further, let $M$ be given a probability measure $\mu$. Then the map induces a probability measure $\nu$ on $N$. The goal is to compute the latter's density with respect to Lebesgue measure, $\frac{d\nu}{d\lambda}$.

Consider the derivative matrix of $\pi$, $d\pi$, of dimension $m \times n$. One can take the generalized Jacobian at $x \in M$ defined by \begin{equation} J_x:= \sqrt{\det (d\pi_x^t d\pi_x)}, \end{equation} where $d\pi_x^t$ denotes the transpose of the matrix $d \pi_x$. Note that $\pi$ being a submersion ensures that $J_x$ is nonzero, and that $\pi^{-1}(y)$ is always a smooth sub-manifold (see Guillemin and Pollack).

Intuitively, a large $J_x$ means a small neighborhood around $x$ will get mapped onto a large neighborhood around $\pi(x)$, hence the density $\frac{d \nu}{d \lambda}$ at $\pi(x)$ should be inversely related to $J_x$. Also larger $\frac{d\mu}{d\lambda}(x)$ obviously gives larger $\frac{d \mu}{d\lambda}$. This reasoning leads to the following conjecture:

\begin{equation} \frac{d \nu}{d\lambda}(y) = \int_{\pi^{-1}(y)} J_x^{-1} \frac{d\mu}{d\lambda}(x) \lambda_{m-n}(dx), \end{equation} where $\lambda_{m-n}$ is the Lebesgue-induced measure on the $(m-n)$-dimensional sub-manifold $\pi^{-1}(y)$, coinciding with the Hausdorff fractal measure.

My question is, is there any textbook/paper that deals with the "obvious" equality above? Or is it an easy consequence of standard results such as the change of variable formula?