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Continuous selections for a class of non-convex multivalued maps. (English) Zbl 0534.28003
Let S and T be compact spaces and let \(\mu_ 0\) be a nonnegative, regular, Borel measure on T. Denote by \(L^ 1(T,Z)\) the Banach space of \(\mu_ 0\)-integrable functions from T into the separable Banach space Z.
A set \(K\subset L^ 1(T,Z)\) is called decomposable, if for all u,\(v\in K\) and A \(\mu_ 0\)-measurable, \(u\cdot \chi_ A+v\cdot \chi_{T\backslash A}\in K\) holds. The main result states, that lower semicontinuity of the map K defined on S with closed and decomposable values in \(L^ 1(T,Z)\) implies the existence of a continuous function \(k:S\to L^ 1(T,Z)\) (called continuous selection), such that \(k(s)\in K(s)\) for all s. Moreover, there exists a countable family of continuous functions \(k_ n:S\to L^ 1(T,Z)\) such that for all \(s\in S K(s)=closure\{k_ n(s):n=1,2,...\}\) holds.
This theorem generalized the theorem of Antosiewicz and Cellina, who proved the existence of a continuous selection for the map in the form \(K(s)=\{u\in L^ 1([a,b],R^ m):u(t)\in F(t,s(t))\quad almost\quad everywhere\quad in\quad [a,b]\}.\) The main result can be generalized in the case, when the map K has decomposable values in Orlicz space.

MSC:
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
54C60 Set-valued maps in general topology
28B05 Vector-valued set functions, measures and integrals
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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