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.^[1][2][3] It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.^[4]

Many classical selection results follow from this theorem^[5] and it is widely used in mathematical economics and optimal control.^[6]

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.^[7]

• Selection theorem

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