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Kuratowski and Ryll-Nardzewski measurable selection theorem

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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

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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 subset of , we have

Then has a selection that is --measurable.[7]

See also

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References

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  1. ^ Aliprantis; Border (2006). Infinite-dimensional analysis. A hitchhiker's guide.
  2. ^ Kechris, Alexander S. (1995). Classical descriptive set theory. Springer-Verlag. ISBN 9780387943749. Theorem (12.13) on page 76.
  3. ^ Srivastava, S.M. (1998). A course on Borel sets. Springer-Verlag. ISBN 9780387984124. Sect. 5.2 "Kuratowski and Ryll-Nardzewski’s theorem".
  4. ^ Kuratowski, K.; Ryll-Nardzewski, C. (1965). "A general theorem on selectors". Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13: 397–403.
  5. ^ Graf, Siegfried (1982), "Selected results on measurable selections", Proceedings of the 10th Winter School on Abstract Analysis, Circolo Matematico di Palermo
  6. ^ 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.
  7. ^ V. I. Bogachev, "Measure Theory" Volume II, page 36.