Does anyone have a reference for the proof of following Theorem:
(Lusin - Novikov) Let X,Y be two standard Borel space and $E \subset X \times Y$ a Borel subset. We assume that $\forall x \in X$ the fiber $\pi^{-1}(x) \cap E$ is countable where $\pi: X \times Y \to X$. Then there is a countable partition $E=\bigcup_{n \in \mathbb{N}}{E_n}$ where $E_n$ is Borel for all n such that $\pi |_{E_n} : E_n \to X$ is injective.
In the literature it is referred to as Lusin-Novikov Theorem, but a google search of this does not reveal any results.
It's in Kechris, Classical Descriptive Set Theory, Theorem 18.10.