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On Carathéodory’s selectors for multifunctions with values in s- contractible spaces. (English) Zbl 0619.28007
Let X be a topological measure space, Y and Z be topological spaces. Let $$F:X\times Y\to Z$$ be such that X-sections are lower semi-continuous and Y-sections are measurable. The question of existence of a Carathéodory selector $$f:X\times Y\to Z$$ is examined, i.e., when there is an $$f:X\times Y\to Z$$ with $$f(x,y)\in F(x,y)$$ for $$(x,y)\in X\times Y;$$ $$f_ x$$ are continuous for all $$x\in X,$$ and $$f^ y$$ are measurable for all $$y\in Y.$$
Lemma 2 establishes a Scorza-Dragoni type property for multifunctions with continuity replaced by lower semi-continuity. It enables to use a selection theorem by Pasicki for multifunctions into a space with his S- convex structure, and a selection theorem by Michael for multifunctions into a space with Michael’s convex structure, to find Caratheódory selections for multifunctions into Z with the corresponding structure.
Reviewer: P.Holicky
MSC:
 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 54C65 Selections in general topology 52A01 Axiomatic and generalized convexity