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On superpositionally measurable multifunctions. (English) Zbl 0705.28003

The paper deals with multifunctions of two variables whose X-sections are continuous and Y-sections are lower measurable with respect to an arbitrary, not necessarily complete, measurable space. It is proved, by using Castaing representation, that in the case of relatively compact values in a metric space and in case the domain being a Cartesian product of a measurable space with a Polish one, the superposition \(F(x,G(y))\) is lower measurable for every lower measurable multifunction G, F a Carathéodory multifunction. The problem of measurability of Carathéodory’s superposition has many consequences in the area of random differential inclusions.
Reviewer: W.Ślȩzak

MSC:

28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
26B40 Representation and superposition of functions
26E25 Set-valued functions
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