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 set-valued function 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 $$ X $$ be a Polish space, $$ \mathcal{B} (X) $$ the Borel &sigma;-algebra of $$ X $$, $$ (\Omega, \mathcal{F}) $$ a measurable space and $$ \psi $$ a multifunction on $$ \Omega$$ taking values in the set of nonempty closed subsets of $$ X $$.

Suppose that $$ \psi $$ is $$ \mathcal{F} $$-weakly measurable, that is, for every open subset $$ U $$ of $$ X $$, we have


 * $$\{\omega : \psi (\omega) \cap U \neq \empty \} \in \mathcal{F}. $$

Then $$ \psi $$ has a selection that is $$ \mathcal{F} $$-$$ \mathcal{B} (X) $$-measurable.