For context I am looking at disjoint components of the Riemann sphere that map to one another injectively by a rational map $R$. i.e. we have the chain $$U_0 \xrightarrow{R} U_1 \xrightarrow{R} U_2 \xrightarrow{R} ...$$ where $R\vert_{U_i}:U_i \rightarrow U_{i+1}$ is injective.
Suppose I have some measureable function $\widetilde{\mu}:U_0 \rightarrow \mathbb{C}$.
I consider the equivalence relation on $\widetilde{\mathbb{C}}:$ $$x \sim y \iff R^n (x)=R^m(y)$$ for some $n,m \in \mathbb{Z}$. (Here $R^n$ denotes the $n-$fold iteration of $R$). Since the $U_i$ are pairwise disjoint and $R\vert_{U_i}$ are injective, at most one point in $U_0$ can be in an equivalence class.
I define the function $\mu:\widetilde{\mathbb{C}} \rightarrow \mathbb{C}$ as follows: for $p\in\widetilde{\mathbb{C}}$, $\mu(p)=0$ if $[p] \cap U_0 = \emptyset$ and $\mu(p)=\widetilde{\mu}(q)$ if $[p]\cap U_0 = \{q\}$.
I want to show that this function defined is also measureable, but I am not entirely sure how. I was thinking of considering $\mu$ on each of the $U_i$ and showing that was measureable... but I don't really know how to go about it.
Any help would be greatly appreciated.