Characterization of L-1-closed decomposable sets in L-infinity

In this paper the set of essentially bounded measurable selections of a mea surable set-valued map is considered. One of the main results states that o nly those subsets of L-infinity that are L-1-closed in L-infinity and decom posable can be represented in this way. The connection between topological and convexity properties of the set-valued map and its L-infinity-selection set is also studied in details, The adjacent and Clarke's tangent cones to L1-closed and decomposable sets and their polar cones are described in ter ms of the corresponding set-valued map as well. (C) 1999 Academic Press.