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 multifunction to have a measurable selection.^[1][2][3] It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.

Many classical selection results follows from this theorem^[4] and it is widely used in mathematical economics and optimal control.^[5]

Statement of the theorem

Let X be a Polish space, ℬ(X) the Borel σ-algebra of X, (Ω, B) a measurable space and Ψ a multifunction on Ω taking values in the set of nonempty closed subsets of X.

Suppose that Ψ is B-weakly measurable, that is, for every open set U of X, we have

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

Then Ψ has a selection that is B-ℬ(X)-measurable.^[6]

See also

• List of selection theorems

References

1. Aliprantis; Border (2006). Infinite-dimensional analysis. A hitchhiker's guide.
2. Kechris, Alexander S. (1995). Classical descriptive set theory. Springer-Verlag. Theorem (12.13) on page 76.
3. Srivastava, S.M. (1998). A course on Borel sets. Springer-Verlag. Sect. 5.2 "Kuratowski and Ryll-Nardzewski’s theorem".
4. Graf, Siegfried (1982), "Selected results on measurable selections" (PDF), Proceedings of the 10th Winter School on Abstract Analysis, Circolo Matematico di Palermo
5. Cascales, Bernardo; Kadets, Vladimir; Rodríguez, José (2010). "Measurability and Selections of Multi-Functions in Banach Spaces" (PDF). Journal of Convex Analysis 17 (1): 229–240. Retrieved 7 April 2015.
6. V. I. Bogachev, "Measure Theory" Volume II, page 36.