Let $f:X \to X$ a locally invertible non-singular* map of the Lebesgue space[1] $(X,\mathcal{B},m)$ and a partition $\{X_i\}$ of $X$ with $m(X_i)<\infty$ and $\mathcal{B}$ is the smallest $\sigma$-algebra containing $\bigcup f^{-n}(X_i)$ which is complete with respect to $m$.
Define
$$(m\circ f)(A):=\sum_{i=0}^{\infty}m(f(A \cap X_i)).$$
We have that $m\ll m\circ f$ so let $g=\frac{dm}{dm\circ f}$ be the Radon-Nikodym derivative.
Let $Y\subset X$ be a measurable subset, we can restrict the map and the measures in this subspace, so that $g_{Y}=\frac{dm_Y}{dm_Y \circ f|_Y}$.
Now I need to show the identity,
$$\log{g_Y}=\sum_{i=0}^{\tau_Y-1}{\log{g\circ f^i}}\tag{1}\label{eq}$$
where $\tau_Y=\inf{\{n\ge1:f^n \in Y\}}$ is the first return time.
A measurable map $f:X\rightarrow X$ on a measure space $(X,\mathcal{B},m)$ is nonsingular if $m(f^{-1}(A))=0$ for all $A\in \mathcal{B}$ such that $m(A) = 0$
My first attempt was to work "backwards" so that I could see how to get to $\eqref{eq}$, but I only managed to reach the expression $$g_Y = \prod_{i=0}^{\tau_Y-1}g\circ f^i.$$
This expression is very similar to an exercise in Viana's book, Foundations of Ergodic Theory. Exercise 9.7.5, but I couldn't solve it.
I believe that by the uniqueness of the Radon–Nikodym derivative, in $Y$ we have to have the equality $g = g_Y$, but I don't know how this can be useful to me.
Does anyone have any idea how to find this equality? And is she known? It seems to me that yes, it is very similar to the definition of entropy....