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. It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.
Many classical selection results follow from this theorem and it is widely used in mathematical economics and optimal control.
Statement of the theorem.
Let formula_1 be a Polish space, formula_2 the Borel σ-algebra of formula_1, formula_4 a measurable space and formula_5 a multifunction on formula_6 taking values in the set of nonempty closed subsets of formula_1. 
Suppose that formula_5 is formula_9-weakly measurable, that is, for every open subset formula_10 of formula_1, we have
formula_12
Then formula_5 has a selection that is formula_9-formula_2-measurable.
References.
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