# Integration of multifunctions with respect to a multimeasure

Brink, Harry Edward (1999-12)

Dissertation (Ph.D)--University of Stellenbosch, 1999.

Thesis

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.

AFRIKAANSE OPSOMMING: Die hoofdoel van hierdie tesis is om die integraal van 'n multifunksie F (waar F vanaf 'n puntversameling T na 'n Banach ruimte X gedefinieer is) met betrekking tot 'n multimaat M (waar M op 'n ring R gedefinieer is en met waardes in 'n Banach ruimte Y) te definieer en dan die eienskappe te ondersoek. Die integrasie van multifunksies met betrekking tot 'n vektormaat is omvattend bestudeer as gevolg van die toepassings wat dit in wiskundige ekonomie het. Daarenteen het Papageorgiou [55], en later Kandilakis [44], integrasie van 'n funksie met betrekking tot 'n multimaat bestudeer. Ons definieer ons integraal in terme van die selektors van die multifunksie F en die selektors van die multimaat M sodat beide bostaande integrale spesiale gevalle is van ons integraal. Die eerste twee hoofstukke dien as 'n inleiding en vorm die grondslag van die werk in die daaropvolgende hoofstukke. In die eerste hoofstuk hersien ons sommige van die basiese definisies en resultate van die teorie van vektormate en meetbare funksies. In die besonder gee ons 'n kort oorsig van die proses waarvolgens 'n vektormaat m, gedefinieer op 'n ring R van deelversamelings van 'n puntversameling T, uitgebrei word na 'n o-ring wat vir R bevat. Hoofstuk 2 word gewy aan die basiese teorie van multifunksies en multimate. Die standaard verwysing vir die gedeelte oor meetbare multifunksies is Maritz [51], wat meetbaarheid van die multifunksie (Definisie 2.1.2) gedefinieer het as die versamelingswaardige weergawe van die meetbaarheid van 'n funksie (Definisie 1.3.5). Ons begin met 'n bespreking van Maritz [51] se uiteensetting van die karakterisering van meetbaarheid van 'n multifunksie in terme van sy grafiek, sy inverse en sy Castaing-voorstelling. Laastens ondersoek ons die meetbaarheid van sekere spesiale multifunksies, naamlik die ekstreempuntmultifunksie en die geslote konvekse omhulsel multifunksie. Die grootste gedeelte van Hoofstuk 2 word gewy aan die teorie van multimate. Deur gebruik te maak van Godet-Thobie [36] definieer ons drie verskillende tipes multimate en bespreek dan die logiese implikasies tussen hulle. Verder skets ons dan ook die bestaan van selektors van 'n multimaat M en bespreek vervolgens die topologiese eienskappe van SM, die klas van alle selektors van M. In die besonder ondersoek ons die voorwaardes wat sal waarborg dat SM # 0 en M(A) = {m(A) / m E SM}. Laastens bestudeer ons oorgangsmultimate, met ander woorde multimate wat geparametriseer word deur elemente van 'n meetbare ruimte. In Hoofstuk 3 bewys ons uitbreidingsresultate vir multimate en oorgangsmultimate. Ons begin deur additiewe versamelingswaardige funksies uit te brei. Ons resultate is volgens die uitbreidingsproses vir vektormate soos in Hoofstuk 1 bespreek. In die hoofresultaat (Stelling 3.1.12) van hierdie hoofstuk bewys ons die versamelingswaardige weergawe van die Caratheodory-Hahn-Kluvanek stelling. In die proses brei ons die ooreenkomstige resultaat (Stelling 3.1.7) van Kandilakis [44] uit na additiewe versamelingswaardige funksies. Ons sluit die hoofstuk af met uitbreidingsresultate vir normale multimate en oorgangsmultimate. In die eerste gedeelte van Hoofstuk 4 hersien ons die bilineere integraal J f(t)m( dt) van 'n funksie f : T - X met betrekking tot 'n vektormaat m : R - Y soos ontwikkel deur Dinculeanu [27]. Die integral J F(t)M(dt) van 'n multifunksie F met betrekking tot 'n multimaat M word dan gedefinieer in terme van J f(t)M(dt). Ons ondersoek dan verder die konveksiteit en kompaktheid van ons integraal en terselfdertyd bewys ons Radon-Nikodym-tipe stellings vir hierdie integraal. Laastens bespreek ons die kommutatiwiteit van die geslote konvekse omhulsel operator en die ekstreempuntoperator met die integraaloperator. Laastens, in die eerste gedeelte van Hoofstuk 5 bestudeer ons die eienskappe van die ruimte van integreerbaar-begrensde meetbare multifunksies. In die besonder bewys ons dat die ruimte van aIle integreerbaar-begrensde, meetbare en konveks- en kompakwaardige multifunksies separabel is. Ons bewys ook die ekwivalensie van ons integraal met die van Debreu [24]. Ons sluit dan die hoofstuk af met 'n ondersoek na die eienskappe van multimate wat gedefinieer word deur digthede en ons bewys die versamelingswaardige weergawe van die Lebesgue-ontbindingstelling.

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