Characterization of L1-Closed Decomposable Sets in L`*

In this paper the set of essentially bounded measurable selections of a measurable set-valued map is considered. One of the main results states that only those subsets of L` that are L1-closed in L` and decomposable can be represented in this way. The connection between topological and convexity properties of the set-valued map and its L`-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 terms of the corresponding set-valued map as well