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The Strict Topology and Compactness in the Space of Measures

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The Strict Topology and Compactness in the Space of Measures Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1965 The trS ict Topology and Compactness in the Space of Measures. John Bligh Conway Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Conway, John Bligh, "The trS ict Topology and Compactness in the Space of Measures." (1965). LSU Historical Dissertations and Theses. 1068. https://digitalcommons.lsu.edu/gradschool_disstheses/1068 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. This dissertation has been microfilmed exactly as received 6 6—7 24 CONWAY, John Bligh, 1939- THE STRICT TOPOLOGY AND COMPACTNESS IN THE SPACE OF MEASURES. Louisiana State University, Ph.D., 1965 Mathematics University Microfilms, Inc., Ann Arbor, Michigan Reproduced with permission of the copyright owner. Further reproduction prohibitedpermission. without THE STRICT TOPOLOGY AND COMPACTNESS IN THE SPACE OF MEASURES A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics John B/^Conway B.S., Loyola University, 196I August, 1965 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENT The author wishes to express his appreciation to Professor Heron S. Collins for his advice and encourage­ ment. This dissertation was written while the author held a National Science Foundation Cooperative Fellowship. Partial support was also furnished by NSF Grant GP 1449, ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS CHAPTER Page ACKNOWLEDGEMENT..... ................... ii INTRODUCTION...... iv I PRELIMINARIES................... •...... K II BASIC PROPERTIES OF THE STRICT TOPOLOGY 17 III THE MACKEY TOPOLOGY ON C(S)^ ............. 37 IV COMPACTNESS AND SEQUENTIAL CONVERGENCE IN THE SPACE M(S)...................... 62 V VECTOR VALUED FUNCTIONS AND MEASURES........ 85 VI UNSOLVED PROBLEMS....................... 105 BIBLIOGRAPHY........................... 107 BIOGRAPHY.............................. 112 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT The strict topology ^ on C(S), the bounded conti­ nuous complex valued functions on the locally compact Hausdorff space 8, was first introduced by R. C. Buck. Ihe primary concern of this dissertation is the relation­ ship between 0(3)^ and its adjoint M(S), the bounded Radon measures. In particular, when is C(S)^ a Mackey space? From our answer to this question we are able to prove several theorems on various types of compactness and convergence in M(S). Ihe first chapter contains preliminary material with virtually no proofs. Chapter II contains the basic pro­ perties of the strict topology. Some of these results are known and some of the old theorems are presented with new proofs. In particular, we prove Buck's result that C(S)^ * = M(S) and we calculate a basis for the ^ - equicontinuous sets in M(S). The third and fourth chapters contain the heart of this work. Chapter III begins with necessary and sufficient conditions for -equicontinuity in M(S) and a proof that ) is a strong Mackey space. Using these two results we prove the principal theorem of this chapter. This result is that if S is paracompact then every ^ -weak * countably compact subset of M(S) is ^ -equicontinuous; iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. consequently, C(S)^ is a strong Mackey space. Also we show that if S is the space of ordinal numbers less than the first uncountable then C(S^ is not a Mackey space. Chapter III concludes with a characterization of the closed subspaces of ( ) which are Mackey spaces, and a proof that (h “,^ ) is not a Mackey space. After a few preliminary lemmas the fourth chapter begins with some -results concerning ^ -equicontinuity. For example, every ^ -weak * compact subset of positive measures is ^0 -equicontinuous. If S is metrizable then ^ -weak. * sequential, countable, and conditional com­ pactness are all equivalent to ^ -equicontinuity. Then we show how the concept of ^ -equicontinuity and the main theorem of Chapter III can be combined to generalize, improve, and give new proofs of some theorems of J. DieudonnC on various types of sequential convergence in M(S). Finally we use all these facts to prove a weak compactness theorem of A. Grothendieck. Chapter V contains generalizations of the preceding results to vector valued measures and functions. Also we characterize the weakly compact operators from a Banach space E into M(S). Using this, we show that a weakly compact operator from a sub space of E into M(S) can be extended to a weakly compact operator of E into M(S). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INTRODUCTION The strict topology ^ on C(S), the bounded continuous complex valued functions on the locally compact space S, was first introduced by R, C, Buck [10,11,12], It has also been studied by Glicksberg [l8] and Wells [36], This topology has been used in the study of various problems in spectral synthesis (Herz [22]), spaces of bounded analytic functions (Shields and Rubel [31,32]), and multipliers of Banach algebras (Wells [35] and Wang [34]). In spite of these successful applications, there has as yet been no detailed investigation of the relationship between 0(8)^^ and its adjoint space M(S), In particular, it is not known whether or not C(8)yg is a Mackey space (a question asked by Buck [12]). It is one of the purposes of this dissertation to begin such an investigation. Hie existence and description of the Mackey topology, the strongest topology yielding a given adjoint space, is a natural object for consideration, Œîiis topology can be described for general locally convex spaces, and there are several conditions (e.g. metric) which imply that a topology is a Mackey topology. Nevertheless, for a parti­ cular locally convex space E which is not a priori a Mackey space, the question: of whether or not E is a Mackey space may be extremely difficult. Moreover, the general 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. description of the Mackey topology may be totally unsuitable for a concrete space. Indeed, the author knows of no examples in which a space with an intrinsically defined topology is shown to be a Mackey space (unless it has some other formally stronger property like metric; etc.). However, if S is a paracompact non-compact space then C(S)^ is a Mackey space which is not metric, barrelled, or bornological (Theorem 3,7). Also we give necessary and sufficient conditions for a closed subspace of ^ ) to be a Mackey space, and we show that (h “, ^ ) is not a Mackey space. In the pro­ cess we show that ( **> ^ ) has closed subspaces which are not Mackey spaces. : Tbe second purpose of this paper is to present the proof of several compactness and sequential convergence criteria for M(S). There is a wealth of literature on this matter. In fact, in addition to some results of our own on these subjects, we succeed in applying our Theorem 3.7 to an investigation of the results of J. Dieudonné [13]. Our work here consists of generalizations to locally compact spaces, improvements, and the elimination of many of Dieudonne ' s sorguments through the use of Theorem 3.7. Every effort has been made to make the reader's Job painless. In addition to the inclusion of detail, which to some may seem tedious, we have also added an index of symbols at the end of Chapter I. All theorems, corollaries. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and lemmas in a given chapter have been numbered con­ secutively, Also Theorem x.y means theorem number y in chapter x. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I PRELIMINARIES In this chapter we have endeavored to present a variety of results and facts which we hope will facilitate the reading of this dissertation. Some of the terms we will use can be found in the literature with a meaning different from ours. For this reason we advise the reader to being with at least a cursory reading of this section. Topology and continuous functions. Unless otherwise stated the topological notions used will be that of Kelley [24]. However, there are some notable exceptions. To avoid cumbersome phraseology we shall adapt the following terminology. If X is a topo­ logical space and A a subset of X then A is countably compact if and only if every sequence in A has a cluster point in X (not necessarily in A). Also A is sequentially compact if and only if every sequence in A has a subsequence which converges to some point of X. Finally A is con­ ditionally compact if and only if A’ (the closure of A ) is compact. Throughout this work S will always denote a locally compact Hausdorff space, and int A the interior of the set A. THEOREM 1.1. ([6,p. 107]) Die space S is paracompact 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. if and only if S is the union of a pairwise disjoint collection of open and closed CT -compact subsets of S, In particular, the above theorem says that if S is <r-compact or a topological group then S is paracompact, LetJ\^ be the space of ordinal numbers less than the first uncountable ordinal-fl, with the order topology.
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