On selection theorems with decomposable values (Q1590117)

The authors study the existence of continuous selections for multivalued mappings from a paracompact space \(X\) into \(L_{1}(T,E)\). They introduce the class of dispersibly convex subsets of \(L_{1}(T,E)\), which contains balls and decomposable sets. This class is closed under intersections, linear combinations and the closure operation. Then they introduce the notion of a dispersible multivalued mapping from \(X\) to \(L_{1}(T,E)\). The main result of the paper says that if \((T,\mathcal{F},\mu)\) is a separable and nonatomic probability space, then every dispersible and closed-valued mapping from \(X\) to \(L_{1}(T,E)\) has a continuous selection. The proof follows the classical scheme of the proof of the Michael convex-valued selection theorem. Since every lower semicontinuous mapping with decomposable values is dispersible, the paper provides a new proof of the Fryszkowski selection theorem [\textit{A. Fryszkowski}, Stud. Math. 76, 163-174 (1983; Zbl 0534.28003); see also \textit{A. Bressan} and \textit{G. Colombo}, ibid. 90, No. 1, 69-86 (1988; Zbl 0677.54013)], without use of the Lyapunov theorem on the convexity of the set of values of vector measures.