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

Suppose that ${\displaystyle \psi }$ is ${\displaystyle {\mathcal {F}}}$-weakly measurable, that is, for every open subset ${\displaystyle U}$ of ${\displaystyle X}$, we have

${\displaystyle \{\omega$ :\psi (\omega )\cap U\neq \emptyset \}\in {\mathcal {F}}.}

Then ${\displaystyle \psi }$ has a selection that is ${\displaystyle {\mathcal {F}}}$-${\displaystyle {\mathcal {B}}(X)}$-measurable.^[7]

See also

• Selection theorem

References

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.