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 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 be standard Borel spaces and a subset that is measurable with respect to the analytic sets. Then there exists a measurable function such that, for all, if and only if.
An application of the theorem is that, given any measurable function, there exists a universally measurable function such that for all.