Kuratowski and Ryll-Nardzewski measurable selection theorem explained

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 and it is widely used in mathematical economics and optimal control.^[5]

Statement of the theorem

Let

X be a Polish space, l{B} (X) the Borel σ-algebra of X , (\Omega, l{F}) a measurable space and \psi a multifunction on \Omega taking values in the set of nonempty closed subsets of X .

Suppose that

\psi is l{F} -weakly measurable, that is, for every open subset U of X , we have\{\omega : \psi (\omega) \cap U ≠ \empty \} \in l{F}.

Then

\psi has a selection that is l{F} -l{B} (X) -measurable.^[6]

See also

• Selection theorem

Notes and References

1. Book: Aliprantis. Border. Infinite-dimensional analysis. A hitchhiker's guide.. 2006.
2. Book: Kechris, Alexander S.. Classical descriptive set theory. registration. Springer-Verlag. 1995. 9780387943749 . Theorem (12.13) on page 76.
3. Book: Srivastava, S.M.. A course on Borel sets. Springer-Verlag. 1998. 9780387984124 . Sect. 5.2 "Kuratowski and Ryll-Nardzewski’s theorem".
4. Kuratowski. K.. Ryll-Nardzewski. C.. A general theorem on selectors. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.. 13. 1965. 397–403.
5. Cascales . Bernardo . Kadets . Vladimir . Rodríguez . José . 2010 . Measurability and Selections of Multi-Functions in Banach Spaces . Journal of Convex Analysis . 17 . 1 . 229–240 . 28 June 2018.
6. V. I. Bogachev, "Measure Theory" Volume II, page 36.