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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

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  • $\begingroup$ Sure, You are right: I forgot to add that $X,Y$ are standard Borel spaces. I'm sorry $\endgroup$ – truebaran Oct 20 '18 at 13:33
  • $\begingroup$ What about taking Y to be the reals R, and X to be a disjoint union of R with R, and p the identity on one copy of R, and on the other copy some example of a measurable map from R to R that doesn't have a measurable image. Then take n = 2. $\endgroup$ – Lorenzo Najt Oct 20 '18 at 14:25
  • $\begingroup$ @Lorenzo is X a complete metric space? $\endgroup$ – Tim kinsella Oct 20 '18 at 14:30
  • $\begingroup$ I guess it doesn't matter. You can just do what you said on two halves of $\mathbb{R}$. Provided there exists an injective measurable map whose image isn't measurable, as you say. $\endgroup$ – Tim kinsella Oct 20 '18 at 14:48
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As in the answer I provided in MO, this follows from Exercise 18.15 in Kechris' Classical Descriptive Set Theory, which states (adapted to your notation) what you need that but for a subset $P\subseteq Y\times X$ with countable sections $P_y$ (you can take $P:=(\mathrm{graph}(p))^{-1}$).

For $n=0,\dots,\infty$, 18.15 says that $Y_n$ is Borel and there are Borel functions $f_i^{(n)}:Y_n\to X$ with disjoint graphs such that for $x\in Y_n$, $p^{-1}(x) = \{f_i^{(n)}(x) : i <n\}$. This exercise follows from the Lusin-Novikov Theorem (18.10 in the same book) that states the projection of such a $P$ must be Borel in $X$.

In particular, there are no injective, Borel measurable maps between complete, separable metric spaces, with non Borel image.

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