Kuratowski and Ryll-Nardzewski measurable selection theorem

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.

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.