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 edit
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 .
References edit
- Kechris, Alexander (1995), Classical descriptive set theory, Springer-Verlag.
- von Neumann, John (1949), "On rings of operators, Reduction theory", Ann. Math., 50: 448–451.