Lusin–Novikov — a Borel set with countable sections is a countable disjoint union of Borel graphs

Let $P \subseteq X \times Y$ be a Borel set whose every vertical section $P_x = \{y \mid (x, y) \in P\}$ is countable. Then $P$ is the union of a sequence $G_0, G_1, \dots$ of pairwise disjoint Borel sets, each of which is a graph: if $(x, y)$ and $(x, y')$ both lie in $G_n$, then $y = y'$ (the first projection is injective on $G_n$).

Here $X$ and $Y$ are standard Borel spaces (StandardBorelSpace), “Borel” means measurable for their σ-algebras, and $X \times Y$ carries the product σ-algebra. "Countable" includes finite.

Kechris states the Lusin–Novikov theorem as Theorem 18.10 (p. 123): for a Borel $P \subseteq X \times Y$ all of whose sections $P_x$ are countable, $P$ has a Borel uniformization, its projection is Borel, and $P$ is a countable union of Borel graphs. This is the last of the three claims, with the graphs in addition pairwise disjoint. The proof here does not follow Kechris: it is Shinko's argument with σ-ideals (2024), which proves that a continuous map of Polish spaces with countable fibres is injective on each of countably many Borel sets covering its domain (LusinNovikov.exists_injOn_cover_of_countable_fibers), and applies it to the projection $P \to X$.