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.[1][2][3] It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.[4] Many classical selection results follow from this theorem[5] and it is widely used in mathematical economics and optimal control.[6]
Statement of the theorem
Let [math]\displaystyle{ X }[/math] be a Polish space, [math]\displaystyle{ \mathcal{B} (X) }[/math] the Borel σ-algebra of [math]\displaystyle{ X }[/math], [math]\displaystyle{ (\Omega, \mathcal{F}) }[/math] a measurable space and [math]\displaystyle{ \psi }[/math] a multifunction on [math]\displaystyle{ \Omega }[/math] taking values in the set of nonempty closed subsets of [math]\displaystyle{ X }[/math].
Suppose that [math]\displaystyle{ \psi }[/math] is [math]\displaystyle{ \mathcal{F} }[/math]-weakly measurable, that is, for every open subset [math]\displaystyle{ U }[/math] of [math]\displaystyle{ X }[/math], we have
- [math]\displaystyle{ \{\omega : \psi (\omega) \cap U \neq \empty \} \in \mathcal{F}. }[/math]
Then [math]\displaystyle{ \psi }[/math] has a selection that is [math]\displaystyle{ \mathcal{F} }[/math]-[math]\displaystyle{ \mathcal{B} (X) }[/math]-measurable.[7]
See also
References
- ↑ Aliprantis; Border (2006). Infinite-dimensional analysis. A hitchhiker's guide..
- ↑ Kechris, Alexander S. (1995). Classical descriptive set theory. Springer-Verlag. ISBN 9780387943749. https://archive.org/details/classicaldescrip0000kech. Theorem (12.13) on page 76.
- ↑ Srivastava, S.M. (1998). A course on Borel sets. Springer-Verlag. ISBN 9780387984124. https://archive.org/details/springer_10.1007-978-0-387-22767-2. Sect. 5.2 "Kuratowski and Ryll-Nardzewski’s theorem".
- ↑ Kuratowski, K.; Ryll-Nardzewski, C. (1965). "A general theorem on selectors". Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13: 397–403.
- ↑ Graf, Siegfried (1982), "Selected results on measurable selections", Proceedings of the 10th Winter School on Abstract Analysis, Circolo Matematico di Palermo, http://dml.cz/dmlcz/701265
- ↑ Cascales, Bernardo; Kadets, Vladimir; Rodríguez, José (2010). "Measurability and Selections of Multi-Functions in Banach Spaces". Journal of Convex Analysis 17 (1): 229–240. http://www.um.es/beca/papers/MeasSelections.pdf. Retrieved 28 June 2018.
- ↑ V. I. Bogachev, "Measure Theory" Volume II, page 36.
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