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 σ-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

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

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

See also

• Selection theorem