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.

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

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

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

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

• Selection theorem