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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, μ- a nonatomic probability measure on a space T, and L 1 - the Banach space of μ-integrable functions u: TE. A set KL 1 is decomposable if u·χ A +v·χ TA K for any μ-measurable set AT and all u,vK. 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

MSC:
54C65Continuous selections
54C20Extension of maps on topological spaces