# 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

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

## See also

## References

- ↑ Aliprantis; Border (2006).
*Infinite-dimensional analysis. A hitchhiker's guide*. - ↑ Kechris, Alexander S. (1995).
*Classical descriptive set theory*. Springer-Verlag. Theorem (12.13) on page 76. - ↑ Srivastava, S.M. (1998).
*A course on Borel sets*. Springer-Verlag. Sect. 5.2 "Kuratowski and Ryll-Nardzewski’s theorem". - ↑ Graf, Siegfried (1982), "Selected results on measurable selections" (PDF),
*Proceedings of the 10th Winter School on Abstract Analysis*, Circolo Matematico di Palermo - ↑ 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. - ↑ V. I. Bogachev, "Measure Theory" Volume II, page 36.