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 {\displaystyle X} be a Polish space, B ( X ) {\displaystyle {\mathcal {B}}(X)} the Borel σ-algebra of X {\displaystyle X} , ( Ω , F ) {\displaystyle (\Omega ,{\mathcal {F}})} a measurable space and ψ {\displaystyle \psi } a multifunction on Ω {\displaystyle \Omega } taking values in the set of nonempty closed subsets of X {\displaystyle X} . Suppose that ψ {\displaystyle \psi } is F {\displaystyle {\mathcal {F}}} -weakly measurable, that is, for every open subset U {\displaystyle U} of X {\displaystyle X} , we have
{ ω : ψ ( ω ) ∩ U ≠ ∅ } ∈ F . {\displaystyle \{\omega :\psi (\omega )\cap U\neq \emptyset \}\in {\mathcal {F}}.}
Then ψ {\displaystyle \psi } has a selection that is F {\displaystyle {\mathcal {F}}} - B ( X ) {\displaystyle {\mathcal {B}}(X)} -measurable.
See also Selection theorem
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