Theory of Charges: A Study of Finitely Additive Measures PDF
Preview Theory of Charges: A Study of Finitely Additive Measures
Theory of Charges This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUEELI LENBERAGND HYMANBA SS A list of recent titles in this series appears at the end of this volume. Theory of Charges A Study of Finitely Additive Measures K. P. S. Bhaskara Rao Indian Statistical Institute, Calcutta, India M. Bhaskara Rao University of Sheffield, UK 1983 Academic Press A Subsidiary of Harcourt Brace Jovanovich, Publishers London New York Paris San Diego San Francisco SHo Paulo Sydney Tokyo Toronto ACADEMIC PRESS INC. (LONDON) LTD 24/28 Oval Road, London NW1 7DX United States Edition published by ACADEMIC PRESS INC. 111 Fifth Avenue, New York, New York 10003 Copyright @ 1983 by ACADEMIC PRESS INC. (LONDON) LTD All Rights Reserved No part of this book may be reproduced in any form by photostat, microfilm, or any other means without written permission from the publishers British Library Cataloguing in Publication Data Bhaskara Rao, K.P.S. Theory of Charges-(Pure and Applied Mathematics) 1. Algebraic Topology I. Title 11. Bhaskara Rao, M. 111. Series 514' 2 QA612 ISBN 0-12-095780-9 Typeset and printed by J. W. Arrowsmith Ltd Foreword Many years ago, S. Bochner remarked to me that, contrary to popular mathematical opinion, finitely additive measures were more interesting, more difficult to handle, and perhaps more important than countably additive ones. At that time, I held the popular point of view, but since then I have come around to Bochner’s opinion. Apparently, many other mathematicians have also done so, as is indicated by the large number of papers listed in the bibliography of this book. I, for one, had not realized how much research had been done on finitely additive measures, at least partly because the material is scattered in isolated papers. The authors have done the mathematical community a real service by providing easy access to this research (to which they themselves have made significant contributions). This service is all the greater in that not only is the material that they cover interesting in itself, but the presentation is very clear and is enlivened with many illustrative examples. Two especially valuable features of their work are an annotated bibliography and a section of notes and comments. But perhaps the most valuable feature of the work to the working measure theorist or functional analyst lies in exhibiting clearly where countable additivity of a measure is used, and what can and what cannot be done without it. Roughly speaking, without countable additivity most of the measure theoretic examples are “counter”, but a great deal of functional analysis can be done-with more work! No one book in an area as large as this can do justice to all the material that deserves coverage, and I certainly do not blame the authors for omitting or treating too briefly some topics which I think are important. I only regret the necessity. I understand that the authors expect to write a book on finitely additive probability also, and perhaps they will include some of the topics so omitted. In any case, I look forward to seeing a continuation of the excellent work they have done in this book. December 1982 Dorothy Maharam Stone University of Rochester Rochester, New York This Page Intentionally Left Blank According to S. Bochner, finitely additive measures are more interesting, and perhaps more important, than countably additive ones (see Maharam (1976)).F initely additive measures arise quite naturally in many areas of analysis. Over the years, there has been a sustained growth of activity in finitely additive measures propelled by mathematicians and statisticians. The case for finitely additive probability is put forward strongly by Dubins and Savage (1965) in their book “How to Gamble If You Must”. They refer to de Finetti, who, in a large number of papers published as early as 1930, “has always insisted that countable additivity is not an integral part of the probability concept but is rather in the nature of a regularity hypothesis.” In fact, Dubins and Savage “view countably additive measures much as one views analytic functions-as a particularly important special case.” But not much attention is paid to finitely additive measures in text-books on Measure Theory. (Books on Functional Analysis do a bit better.) One reason could be that countably additive measures are more tractable than finitely additive ones. A need was felt to have a book on finitely additive measures which could serve as a reference book as well as a text-book. Cultivation of our interest in finitely additive measures started ten years ago. Our sustained interest in this area over the years led us to write this book. In this book we have made an attempt to present a systematic and detailed study of finitely additive measures as we understand them, filling in any gaps that we discerned. This study of finitely additive measures as a mathematical object, in many of its manifestations, is like a study that a botanist would carry on a particular plant, or that a zoologist would launch on a particular species of mammals, or that a sociologist would initiate about a certain tribe, delving deep into various facets of the subject of interest. We look at the finitely additive measure (i) as a single entity (extension, nonatomicity and purity); (ii) in relation to another of its own kind (absolute continuity and singularity); (iii) in an introspective mood (decomposition theorems); (iv) as a member of a community (Nikodym theorem and Vitali-Hahn-Saks theorem); (v) as a member of a community in motion (weak convergence); (vi) in interaction with objects of different ... Vlll PREFACE kind (integration); (vii) in association with related external communities (Vp-s paces); (viii) and its behaviour in external environment (range); (ix) in its internal environment (lifting). Measure Theory (The Study of Countably Additive Measures) is an integral part of this wider study and the contrast between finite additivity and countable additivity is brought into sharp focus at various junctures in this work. This book contains a good number of examples illustrating various aspects of finitely additive measures. A special feature of this book is the Selected Annotated Bibliography provided at the end of the book listing research papers we have come across in our pursuit of finitely additive measures. We hope that this book serves practising analysts well and stimulates further research. K.P.S. Bhaskara Rao gave a series of lectures on some of the topics covered in this book at the University of Lecce (Italy) in 1980 and at the University of Naples in 1981. He acknowledges gratefully the help given by these universities in making the visits possible. We also thank the Indian Statistical Institute for rendering help in making reciprocal visits of the authors possible in connection with this work. Finally, a word of appreciation and gratitude to Surekha for her monu- mental patience in putting up with one of the most taxing and demanding spouses while this work was in progress. We also thank B. R. Marepalli for typing the entire manuscript. December 1982 K.P.S. Bhaskara Rao Calcutta M. Bhaskara Rao Sheffield Contents Foreword V Preface vii CHAPTER 1 PRELIMINARIES 1 1.1 Classes of sets 1 1.2 Set theoretical concepts 13 1.3 Topological concepts 15 1.4 Boolean algebras 18 1.5 Functional analytic concepts 23 CHAPTER 2 CHARGES 35 2.1 Basic concepts 35 2.2 The space of all bounded charges, ba(Q9) 43 2.3 Measures 47 2.4 The space of all bounded measures, ca(R, 9) 50 2.5 Jordan Decomposition theorem 52 2.6 Hahn Decomposition theorem 56 CHAPTER 3 EXTENSIONS OF CHARGES 58 3.1 Real valued set functions and induced functionals 58 3.2 Real partial charges and their extensions 64 3.3 Extension procedure of Eos and Marczewski 70 3.4 Extension of partial charges in the general case 76 3.5 Miscellaneous extensions 78 3.6 Common extensions 82 CHAPTER 4 INTEGRATION 85 4.1 Total variation and outer charges 85 4.2 Null sets and null functions 87 4.3 Hazy convergence 92 4.4 D-integral 96 4.5 S-integral 115 4.6 L,- spaces 121 4.7 ba(O,9) as a dual space 133 CHAPTER 5 NONATOMIC CHARGES 141 5.1 Basic concepts 141 5.2 Sobczyk-Hammer Decomposition theorem 144 5.3 Existence of nonatomic charges 150 5.4 Denseness 156
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