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, $\mathcal (X)$ the Borel σ-algebra of $X$, $(\Omega, \mathcal)$ a measurable space and $\psi$ a multifunction on $\Omega$ taking values in the set of nonempty closed subsets of $X$.

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

:$\ \in \mathcal.$

Then $\psi$ has a selection that is $\mathcal$-$\mathcal (X)$-measurable.V. I. Bogachev
"Measure Theory"
Volume II, page 36.

See also

*Selection theorem

References

Descriptive set theory
Theorems in functional analysis
Theorems in measure theory

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