On page 186, the authors are discussing that if $f:M\rightarrow M$ is a non-invertible smooth map on a manifold $M$, $\Omega$ is a volume form and $\rho:M\rightarrow \mathbb{R}^+$ is a density so that the measure $\mu(A) := \int_A \rho.\Omega$ defines an $f$-invariant measure, then
$$ \rho(x) = \sum_{y \in f^{-1}(x)} \dfrac{\rho(y)}{\mathrm{Jac}(f)(y)}. $$
I was wondering why this should be true. I was guessing that probably you can still define something like a measure theoretic version of the pull back of the volume form for non-invertible smooth maps. Indeed, if $M$ is compact, almost every point has finitely many pre-images by Sard's and Regular value thoerems. Therefore, via summing up sum stuff over probably finitely many pre-images we should be able to get what we want via a few applications of the change of variables theorem. However, I wasn't able to define this pull back for some reason. Either if this is or is not the strategy, why is this formula correct?