We present some regularity properties for the set of distributions induced by the measurable selections of a correspondence over a Loeb space, which include closedness, convexity, compactness, purification, and semicontinuity. We also note that all the properties reported in the main theorems are not satisfied by some correspondences on the unit Lebesgue interval.
1996 Academic Press, Inc.
1. Introduction
Correspondences, which are also called multifunctions, set valued maps, and random sets, have been studied extensively and acquired great importance in recent years. The study of measurable correspondences and their selections has applications to a variety of areas. These include optimization, control theory, pattern analysis, stochastic analysis, and mathematical economics. See [5], [11], [18], [21], [27], and [29] for some of the results.
Let T and S be nonempty sets, and P(S) the power set of S. A mapping from T to P(S)&[<] is called a correspondence from T to S. We usually consider correspondences with measure-theoretic or topological structures. Let F be a correspondence from a probability space (T, T, &) to a Polish space X, where T is a _-algbra on T and & a probability measure on (T, T). As usual a Polish space is used to refer a topological space homeomorphic to some complete separable metric space. The Borel _-algebra on X is denoted by B(X). For simplicity, in this paper we shall only work with probability measures which are complete on the corresponding measurable spaces. The correspondence F is said to be measurable if its graph [(t, x) # T_X: x # F(t)] belongs to the product _-algbra article no.