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And the Riesz Representation Theorem ON THE BIDUAL OF JC(X) AND THE RIESZ REPRESENTATION THEOREM by Hermine Engels A Dissertation Submitted to the Faculty of The College of Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Florida Atlantic University Boca Raton, Florida August 1992 ON THE BIDUAL OF K(X) AND THE RIESZ REPRESENTATION THEOREM by Hermine Engels lis dissertation was prepared under the direction of the candidate's thesis advisor, Dr. Helmut Schaefer, Department of Mathematics, and has been approved by the members of his pervisory committee. It was submitted to the faculty of the College of Science and was cepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in athematics. SUPERVISORY COMMITTEE Chairperson, Thesis Adviso 11 ABSTRACT Author: Hermine Engels Title: On the Bidual of /C(X) and the Riesz Representation Theorem Institution: Florida Atlantic University Dissertation Advisor: Dr. Helmut H. Schaefer Degree: Doctor of Philosophy Year: 1992 Locally compact spaces having property (B) are introduced. (B) is the property that each bounded subset of the topological vector space IC(X) is order bounded. It is shown that there exist spaces X exhibiting (B) that are not paracompact. Also discussed are the consequences of this property for the dual and bidual of IC(X) . Both show features resembling those from the Banach space case (X compact). Under (B), for example, the bidual of /C(X) is Riesz isomorphic to IC(Y) for a suitable locally compact space Y. We also introduce doubly bounded Borel functions Bb(X) which are bounded and vanish outside some compact set. Without using the Riesz represe ntation theorem they are imbedded in the bidual of IC(X) , extending the approach of H. H. Schaefer (19][20] for compact spaces to locally co mpact spaces. Further it is shown that the regular Borel measures form a band in the Riesz dual of Bb (X). These results permit to give a topological characteriz ation of regular Borel measures on X, which yields the Riesz represe ntation theorem as well as a distinguishing property of regular Borel measures as fairly immediate conseq uences. Finally, some relations between Baire and Borel measures are cl iscussed. Ill ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. H. H. Schaefer, for the many stimulating discussions we shared on the topics of this dissertation. His patience, enthusiasm, and encouragement were most beneficial to me in completing this work. I am further indebted to Dr. J. Brewer for his help in making my transfer to Florida Atlantic University possible. I am grateful to Dr. S. Locke for his kind help in resolving many administrative difficulties. I would also like to thank the rest of my co mmittee, Dr. C. Lin and Dr. A. Mandell for their time and interest. Special thanks go to my family. Without their understanding, support and love this dissertation would not have been possible. IV TABLE OF CONTENTS ABSTRACT ..... Ill ACKNOWLEDGEMENTS IV INTRODUCTION TERMINOLOGY AND NOTATION 4 CHAPTER THE BIDUAL OF K(X) 5 A. The Space K(X) . 5 l. Preliminaries 5 2. Dedekind Completeness 6 3. Bounded Sets . 8 4. Spaces Having Property (B) 10 B. The Space 9J1(X) 13 1. The Dual of K(X) . 13 2. Lebesgue Property of the Strong Topology 13 3. Weak Compactness of Order Intervals 14 C. The Bidual of K(X) . 15 1. Basic Properties 15 2. Order Continuous Linear Forms in K(X) 16 3. Representation of the Ideal Generated by A:.( X) 18 4. Order Continuous Dual of 9J1(X ) 21 II RADON , BOREL AND BAIRE MEASURES ON LO C ALLY COMPACT SPACES 24 A. Borel and Baire Fun ctions 24 1. Borel Functions 24 2. Baire Functions 26 v 3. Relations between Borel (Baire) Functions on a Locally Compact Space and its One-Point Compactification 28 B. Borel and Baire Measures. Regularity 30 1. Identification of the Duals of C0 (/'() and ry)Lb(X) 30 2. Imbedding of the Borel and Baire Functions into the Bidual of K(X) 32 3. Regularity . 35 C. The Riesz Representation Theorem 39 1. Reg ular Borel Measures and the Riesz Dual of Bb(X) 39 2. Topological Characterization of Regular Borel Measures and the Riesz Representation Theorem 41 3. Baire Measures 43 4. Supplements 46 LITERATURE 49 VITA .... 51 VI INTRODUCTION In his papers on Radon, Baire , and Borel measures on co mpact spaces I and II ([19] [20]), H.H. Schaefer chooses a new approach to illustrate the connections and relations between the set theoretic and fun ctional analytic aspects of the tllPory of intPg ration on compact spaces. The subject of this thesis arose from the id ea to extend this approach to locally compact spaces. The connection between the locally compact case and the compact case is established through the one-point compactifi cation of X ; however, the details of this progra m turned out to be considerably more complicated than one would anticipate. In solvin g this proble m we noticed that for the space K(X) of all continuous, real-valued fun ctions on X with compact support a nd es pecia lly its I idu a l (he re de noted by K(X)) , some signifi cant results ca n be obtain ed . These as well as the preparatory work for C hapter II are contained in C hapter I. In Section A of C hapter l we explore certain propertif' s o f the space K(X) (particularly, order completeness properties ) whi ch are indispensable for the sequel. We also introduce (Definition A.3.1) locally compact spaces having property (B). Property (13) of a lo call y compact space X is the property that each bounded subset of the topological vec tor space K(X) is ord er bounded ; it is well known that every paracompact X has that property. For this, a new proof is given in Theorem A.4.1 ; more importantly, it is shown in A.4.l that ther(' f'X is t. spacf's X exhibiting (D) that are not paracompact. The consequ ences of this property are numerous a nd signifi cant. for example , weak compactness of the order intervals of the dual 9J1(.X) of K( .Y) (13 .3.2) or order continuity of the strong topology ;3(9J1(X) , K(X)) (8.2.1). Moreover, property (B) impa rts a particula rl y transparent stru cture to the bidual K(X) of K(X) ; on the whole, both 9.11( X) and K(X) exhibit Ri esz theoretic features resembling those which are well-know n to hold, and basic for topological measu re theory, whe n X is compact. Perhaps the most striking of these is that unde r (B) , the bidual K(X) is Ri esz ison1orphi c to K(Y) for a suitable locally compact space Y (C.3.l and C.4.3). These qu estions a re investigated in Sec tions Band C. A main result of Section C is stated in Theorem C .l .3. 1!. tnrns out that the order continu­ ous I in ear fo rms on the id eal J of K( .Y) generated by ,\~ (X) are precisely the e le me nts of 9J1( X). This is a surprising res ult (Example C .l.2) whi ch leads us to ma ny other in teresting statements (I.C.3.3, I.C.4.4, II.B.1.3, !LC .l .l) and turns o ut to be essentia l even fo r the new proof of the Riesz representation theore m (II.C.2.2). The second ma in objective of Section l. C is to take a closer look at t he ideal J in K (X) gene ra ted by K(X) . We come up with two results whi ch again a re very helpful in the attempt to discuss Radon , Baire a nd Borel measures o n locall y compact spaces. F irst, J is order dense in the set of all order continuous linear forms in K(X) (C.2.1) and second , there exists a locally compact space Y such t hat J is Ri esz isomorphic with ..\:(Y) (C.3 .1 ). These results permit us to give a characteri zation of the order co ntinuous dual of 9J1(X), whi ch turns out to be Riesz isomo rphic to the ideal of a ll bounded , continuous fun ctions on Y that a re in tegrable against any p. E 9J1 (X). Moreover , the consequences of property (B) (hinted at above) a nd , additionall y, barreled ness of 9J1(X) come into view very clearly now (Theorem C.4.3). If both assumptions ho ld, the bidua l K(Y ) of K(X) shows a ll essentia l features famili a r fr om the Ba nach space case (X compact). Unfortunately, outside of para.compactness of .X no reasonable conditions seem to be know n t hat imply 9J1 (X) to be barreled . Finally, the rather special case of X being discrete is cha racteri zed by refl exivity of either K (X) or 9J1 (X) (C.3.4 a nd C.3.5). C hapter I I sta rts wit.h the definitions and basic properties of Borel a nd Ba ire fun ct ions; for example, equivalent definitions, relations between Borel (Baire) fun ctions o n a locall y compact space and its one-poin t compactification, and id entification of the dual s paces of Co (!\) a nd 9Jlb(X). Very important for the sequel is the introduction of t he doubly bounded Borel fun ctions Bb(X) (Definition A.1.4).
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