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 function. It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.
Many classical selection results follow from this theorem and it is widely used in mathematical economics and optimal control.

Statement of the theorem

Let be a Polish space, the Borel σ-algebra of, a measurable space and a multifunction on taking values in the set of nonempty closed subsets of.
Suppose that is -weakly measurable, that is, for every open set of, we have
Then has a selection that is --measurable.