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]
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]
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]
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]