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 function.[1][2][3] It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.
Many classical selection results follow from this theorem[4] and it is widely used in mathematical economics and optimal control.[5]
Statement of the theorem
Let be a Polish space, the Borel σ-algebra of , a measurable space and a multifunction on taking values in the set of nonempty closed subsets of .
Suppose that is -weakly measurable, that is, for every open set of , we have
Then has a selection that is --measurable.[6]
gollark: They have okay cameras, low power demands, and usable networking capability.
gollark: Personally, if I were to operate a camera like this, I would attain an outdated phone of some sort and just use that.
gollark: It makes no sense. Dimmer lighting should mean longer exposure.
gollark: What? It should be the other way round.
gollark: Any recent computer will have hardware video encoders for at least H.264 also.
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 28 June 2018.
- V. I. Bogachev, "Measure Theory" Volume II, page 36.
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