Jankov–von Neumann uniformization theorem

In descriptive set theory the Jankov–von Neumann uniformization theorem is a result saying that every measurable relation on a pair of standard Borel spaces (with respect to the sigma algebra of analytic sets) admits a measurable section. It is named after V. A. Jankov and John von Neumann. While the axiom of choice guarantees that every relation has a section, this is a stronger conclusion in that it asserts that the section is measurable, and thus "definable" in some sense without using the axiom of choice.

Statement
Let $$X,Y$$ be standard Borel spaces and $$R\subset X\times Y$$ a subset that is measurable with respect to the analytic sets. Then there exists a measurable function $$f:X\to Y$$ such that, for all $$x\in X$$, $$\exists y, R(x,y)$$ if and only if $$R(x,f(x))$$.

An application of the theorem is that, given any measurable function $$g:Y\to X$$, there exists a universally measurable function $$f:g(Y)\subset X\to Y$$ such that $$g(f(x))=x$$ for all $$x\in g(Y)$$.