Kuratowski and Ryll-Nardzewski measurable selection theorem

Let ${\displaystyle X}$ be a Polish space, ${\displaystyle {\mathcal {B}}(X)}$ the Borel σ-algebra of ${\displaystyle X}$, ${\displaystyle (\Omega ,B)}$ 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 ${\displaystyle B}$-weakly measurable, that is, for every open set ${\displaystyle U}$ of ${\displaystyle X}$, we have

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

Then ${\displaystyle \psi }$ has a selection that is ${\displaystyle B}$-${\displaystyle {\mathcal {B}}(X)}$-measurable.[6]

• Selection theorem