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Extensions and selections of maps with decomposable values. (English) Zbl 0677.54013
Let X be a separable metric space, E - a Banach space, \(\mu\)- a nonatomic probability measure on a space T, and \(L^ 1\)- the Banach space of \(\mu\)-integrable functions u: \(T\to E\). A set \(K\subset L^ 1\) is decomposable if \(u\cdot \chi_ A+v\cdot \chi_{T\setminus A}\in K\) for any \(\mu\)-measurable set \(A\subset T\) and all \(u,v\in K\). The property of decomposability is a good substitute for convexity [cf. C. Olech, Proc. Conf. Catanica/Italy 1983, lect. Notes Math. 1091, 193-205 (1984; Zbl 0592.28008)]. Using this property the authors prove analogues of three theorems by Dugundji, Cellina and Michael on extensions and selections of (multivalued) maps.
Reviewer: K.Nikodem

54C65 Selections in general topology
54C20 Extension of maps
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