Bilinear integration of an extreme point multifunction (Q1078347)

This paper opens with a descriptive discussion in which the author gives the reader a flavor of some of the general properties, most of which are known, for the bilinear integration of multifunctions and related topics. The main purpose of this paper is to show that under suitable assumptions the bilinear integrals of a multifunction F and of the corresponding extreme point multifunction \(ext F\) are equal. To be specific, let \({\mathcal C}\) be a ring of subsets of a given nonempty set T which is a countable union of sets of \({\mathcal C}\). Assume that \(m:{\mathcal C}\to {\mathbb{R}}^ p\) has an extended measure, say still m, on the \(\delta\)-ring \(\Sigma (| m|)\) generated by the total variation measure \(| m|\) of m, which is non-atomic; in short assume that \(m:\Sigma (| m|)\to {\mathbb{R}}^ p\) is non-atomic. Suppose that \(F:T\to 2^{R^ n}\) is an \(| m|\)-measurable integrably bounded multifunction such that F(t) is a nonempty compact convex set of \(R^ n\) for every \(t\in T.\) Denote by \(ext F(t)\) the set of extreme points of F(t) for every \(t\in T\). Then, for each \(A\in \Sigma (| m|),\) \[ \int_{A}F(t) dm=\int_{A}ext F(t) dm \] holds. This result generalizes and extends analogous theorems given by several different authors in the last twenty years. The paper ends with several examples which are used to illustrate that the assumptions of the main theorems cannot be weakened.