In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a multifunction to have a measurable selection. It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.
Many classical selection results follows from this theorem and it is widely used in mathematical economics and optimal control.
Statement of the theorem
Let X be a Polish space, ℬ(X) the Borel σ-algebra of X, (Ω, B) a measurable space and Ψ a multifunction on Ω taking values in the set of nonempty closed subsets of X.
Suppose that Ψ is B-weakly measurable, that is, for every open set U of X, we have
Then Ψ has a selection that is B-ℬ(X)-measurable.
References
Kuratowski and Ryll-Nardzewski measurable selection theorem Wikipedia(Text) CC BY-SA