Let $p:X \to Y$ be a measurable surjection and assume that for each $y \in Y$ the set $p^{-1}(y)$ is at most countable. Define $Y_n$ (for $n \in \mathbb{N} \cup \{ \infty \}$) to be the set of those $y \in Y$ for which there are exactly $n$ distinct $x \in X$ such that $p(x)=y$. Is it clear that each $Y_n$ is measurable?
EDIT: I forgot to add, $X,Y$ are assumed to be standard Borel spaces: i.e. sigma algebras are the sigma algebras of Borel set and $X$ and $Y$ are assumed to be complete, separable metric spaces