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# Doctoral Degrees (Mathematical Sciences)

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### Browsing Doctoral Degrees (Mathematical Sciences) by Author "Brink, Harry Edward"

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- ItemIntegration of multifunctions with respect to a multimeasure(Stellenbosch : Stellenbosch University, 1999-12) Brink, Harry Edward; Maritz, P.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
Show more ENGLISH SUMMARY: The main objective of this thesis is to define and investigate the properties of the integral of a multifunction F (where F is from a point set T into a Banach space X) with respect to a multimeasure M (where M is defined on a ring R and with values in a Banach space Y). Integration of multifunctions with respect to a vector measure has been studied extensively because of its applications in mathematical economics. On the other hand, Papageorgiou [55], and later on Kandilakis [44], considered integration of a function with respect to a multimeasure. We define our integral in terms of the selectors of the multifunction F and the selectors of the multimeasure M so that both the above two integrals are special cases of our integral. The first two chapters serve as an introduction and will provide the foundation for work done in the chapters that follow. In the first chapter we recall some of the basic definitions and results of the subject of vector measures and measurable functions. In particular, we give a brief overview of the procedure of extending a vector measure m, defined originally on a ring R of subsets of a point set T, to a o-ring containing R. Chapter 2 is devoted to the basic theory of multifunctions and multimeasures. The standard reference for the section on measurable multifunctions is Maritz [51], who defined measurability of the multifunction (Definition 2.1.2) as the set-valued version of the measurability of a function (Definition 1.3.5). We start by discussing Maritz's [51] exposition of the characterization of measurability of a multifunction in terms of its graph, its inverse and its Castaing representation. Finally, we consider the measurability of some special multifunctions, namely the extreme points multifunction and the closed convex hull multifunction. The better part of Chapter 2 is devoted to the subject of multimeasures. Following Godet-Thobie [36] we define three different types of multi measures and then discuss the logical implications among them. Next we give an outline on the existence of selectors of a multi measure M and we discuss the topological properties of S M, the class of all selectors of M. In particular, we investigate the conditions which will guarantee that SM i- 0 and such that M(A) = {m(A) I m E SM}. Finally, we study transition multimeasures, that is multimeasures parametrized by the elements of a measurable space. In Chapter 3 we are concerned with extension results for multi measures and transition multimeasures. We start by extending additive set-valued set functions. Our results are along the extension procedure for a vector measure as was discussed in Chapter 1. In the main result of this chapter (Theorem 3.1.12) we prove the set-valued version of the Caratheodory-Hahn-Kluvanek theorem. In the process we extend the corresponding result (Theorem 3.1.7) of Kandilakis [44] to additive set-valued set functions. Finally, we prove extension results for normal multimeasures and transition multimeasures. In the first section of Chapter 4 we review the bilinear integral f f (t )m( dt) of a function f : T - X with respect to a vector measure m : R - Y as developed by Dinculeanu [27]. The integral, f F(t)M(dt) of a multifunction F with respect to a multimeasure M is then defined in terms of f f(t)m(dt). We continue by investigating the convexity and compactness of our integral and in the process we also establish Radon-Nikodym-type theorems for our integral. Finally, we discuss the commutativity of the closed convex hull operator and the extreme points operator with the integral operator. Finally, in the first part of Chapter 5 we study the properties of the space of integrably bounded measurable multifunctions. In particular, we prove that the space of integrably bounded, measurable and compact- and convex-valued multifunctions is separable. In addition we also. prove the equivalence of our integral and the integral of Debreu [24]. Finally, we investigate the properties of multi measures defined by densities and we prove the set-valued version of the Lebesgue decomposition theorem.Show more