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 [math]\displaystyle{ X,Y }[/math] be standard Borel spaces and [math]\displaystyle{ R\subset X\times Y }[/math] a subset that is measurable with respect to the analytic sets. Then there exists a measurable function [math]\displaystyle{ f:X\to Y }[/math] such that, for all [math]\displaystyle{ x\in X }[/math], [math]\displaystyle{ \exists y, R(x,y) }[/math] if and only if [math]\displaystyle{ R(x,f(x)) }[/math].

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

References

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