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? 



Reference request for a result regarding density of induced probability measure under a submersion
Answering my own question: Corollary 1.4 of this online note seems to address exactly the problem: <http://www3.nd.edu/~lnicolae/Coarea.pdf>


As expected it is intimately related to the co-area formula (Theorem 1.3). The latter is proved using Fubini's theorem locally, partition of unity, and implicit function theorem. 


