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Topological Measure Theory, with Applications to Probability

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Topological Measure Theory, with Applications to Probability TOPOLOGICAL MEASURE THEORY, WITH APPLICATIONS TO PROBABILITY by David Bruce Pollard A thesis submitted for the degree of Doctor of Philosophy of the Australian National University June, 1976 (ii) To Camilla and Flemming Tops^e (iii) CONTENTS ABSTRACT ............................................................ (iv) PREFACE (including Statement of Originality) ......... (v) SYMBOLS AND TERMS ................................................... (vii) CHAPTER ONE: Introduction §1. Topology and measure .......................... 1 CHAPTER TWO: Integral representation theorems §2. Construction of measures ...................... 6 §3. Assumptions required for integral representations 12 §4. The largest measure dominated by a linear functional .................................. 16 §5. Finitely additive integral representations 20 §6. o- and t-additive integral representations 25 §7. The Alexandroff-Le Cam theorem .............. 34 §8. Further remarks .............................. 38 CHAPTER THREE: Weak convergence and compactness §9. Weak convergence theory: introduction 40 §10. Weak convergence and compactness for more general spaces ......................... ......... 42 §11. Historical perspective ...................... 53 CHAPTER FOUR: T-additive measures §12. The need for T-additivity .................. 57 §13. Uniform spaces 61 §14. Operations preserving T-additivity ......... 64 §15. Weak convergence of T-measures 68 §16. Induced weak convergence ...................... 79 §17. Infinite T-measures .......................... 86 CHAPTER FIVE: Random measures on locally compact spaces §18. Introduction ........ ......... 88 §19. Existence of random measures 90 §20. Weak convergence of random measures 95 §21. More on weak convergence ......... 98 CHAPTER SIX: Markov chains on topological spaces §22. Markov chains on abstract spaces ......... 102 §23. Strongly continuous chains ............ 107 §24. Weakly continuous chains on completely regular 110 spaces .. .ü •• o. .o .. o * •. .. BIBLIOGRAPHY 115 (iv) ABSTRACT The work carried out for this thesis was motivated by a belief that the methods of topological measure theory could be more widely applied in the theory of probability. As my introduction to the subject was through the field of weak convergence, much of the thesis has developed out of the study of problems in that area; but this has often also involved consideration of the topological, measure theoretic and functional analytic background material. For example, the integral representation theory of Chapter 2 grew out of the study of relative compactness in the topology of weak convergence; the latter is the subject of Chapter 3. But the results of Chapter 2 are also of interest in their own right as they form the basis of a unified approach to Riesz type integral representation theorems. While these investigations were being carried out it became apparent that one particular concept has a most important part to play in topological measure theory. This so-called T-additivity property is examined in Chapter 4. It constitutes an intermediate stage in the progression from countable additivity to the stronger Radon measure concept. It often seems to be the minimal condition for compatibility between the topological and measure theoretic structures; this view is supported by the results of Chapter 4. The last two chapters contain some of the other applications of this research to probability theory. In Chapter 5 problems related to the existence and weak convergence of random measures on locally compact spaces are considered; and in Chapter 6 some aspects of the theory of Markov chains on topological state spaces are discussed. Weak convergence arguments are prominent in both chapters. (v) PREFACE During the three years of my Ph.D. studies I have received assistance from many sources. First my thanks must go to my supervisors Dr C.C. Heyde and Professor P.A.P. Moran. Through advice and discussion they have helped and influenced me in many ways. The material in Chapter 2 is based mainly on a paper (Pollard and Tops^e (1975)) prepared during my visit to the Mathematical Institute of Copenhagen University from September 1973 until August 1974; this stay was generously supported by the Danish Natural Science Research Council. My thanks go to the members of that Institute for their hospitality. I am especially grateful to Dr F. Tops^e, from whom I have learnt a lot. We contributed equally to the joint paper just mentioned. Chapter 3 is an expansion of the revised version (Pollard (1975)) of a paper drafted before the Copenhagen visit. Some of the results in Section 15 (plus extensions) are to be submitted for publication. Section 16 and most of Chapter 5 are based on an already submitted paper (Pollard (1977)). This paper grew out of discussions with Dr G.M. Laslett on some problems in the theory of point processes. The stimulating conversations which I had with him on this and other topics are greatly appreciated. The material of Chapter 6 is mainly taken from a pair of joint papers with Dr R.L. Tweedie (Pollard and Tweedie (1975, 1976)). We shared the development of the topological and measure theoretic aspects of these papers equally, but Dr Tweedie contributed most of the Markov chain theory. He also spent much time patiently explaining many things to m e , offered a lot of very useful advice and helped with the tedious task of proof reading. My thanks to him for all this. My contributions to our other paper referred to in Chapter 6 (Laslett, Pollard and Tweedie (1977)) were mainly concerned with the applications of the weak continuity concept. It was very important to me to be able to work in a stimulating and friendly atmosphere. For this opportunity I am indebted to the Australian National University for financial support. I wish to thank all the members of the Department of Statistics, IAS at that university, for advice and assistance freely given; I sought these often. Two members of this (vi) department deserve special mention. I want particularly to thank Mrs B. Cranston for her help in preparing this thesis, as well as for her practical assistance and cheerful encouragement on numerous occasions. (Visiting) Professor J.A. Hartigan not only patiently endured my explanations of the material of Chapter 2 but also made a lot of helpful criticisms. I had many an interesting conversation with him. Thanks too to Professor T.P. Speed and Dr R.K. Milne for their hospitality and useful remarks during my visit to the Mathematics Department, University of Western Australia in February 1976. Finally I wish to express my gratitude to Mrs B. Geary for her excellent typing, and to my dear wife Sheridan for her love and understanding (and help with the proof-reading). A A A Where I have consciously borrowed results from other authors the appropriate references have been made; sometimes though I have regarded the material as part of the folklore of the subject. But to the best of my knowledge, and with these reservations, I can claim this thesis to be the product of my own original research. (vii) SYMBOLS AND TERMS Some symbols are used with several different meanings; and sometimes different symbols refer to the same object. References to page x are given as: p. x. A ° complement of the se t A A closure of the set A A p. 102 A e p. 77 A\B set theoretic difference A/SB symmetric difference of sets A1 p. 13 A2 p. 13 A3 p. 14 A4 p . 14 , . A5 p . 14 A5 ’ p. 14 A (K) p. 7 atom p. 59 a. s . almost surely BOO P. 7 p. 89 B o B p. 102 B+ p. 103 8i p. 103, p. 104 p. 105, p. 106 B(X) p. 107 C p. 13, p. 88 c+ p. 88 C(X) the space of bounded continuous real functions on X 8 C> o i—i p. 81 Sß the space of bounded uniformly continuous real V cuw functions on X compact paving p. 30 completely Hausdorff p. 33 completely regular p. 32 continuous (measure) p . 59 (viii) convolution (of measures) p . 67 V p. 44, p. 91 D[ 0, «) p . 84 A symmetric difference (of sets) 3A boundary of the set A df the set of discontinuities of the function f p. 73 8 VT/ , £ Jf domination (of a linear p . 16 functional) domination (of a paving) p . 50 downward filtering for every AA^ in the class there (class of sets) exists an A c A n A , with A in d “ J. Z d the class £ probability measure concentrated at the X point x entourage p. 61 equicontinuous p. 49 exhaustion ("K exhausts p. 21 T on CM) F(JC) p. 7 fidi finite dimensional distribution; P* 90 filtering see "upward (downward) filtering" G(K) p. 7 0 r (x, 4) p. 103 indicator function of the set A h intC4) interior of the set A inner integral p. 15 K p. 7 p. 28 Ka K p. 28 T /(-regular finitely p. 8 additive /(-regular a-additive p. 8 /(-regular T-additive p. 8 /(-tight p. 8 (ix) L p . 43 UV) p . 44 L(K) p . 47 L-set p. 106 l.s.c. lower semi-continuous A/+(X, t) p. 50 M + (X, t ) p. 52 Mt , Mt (X) p. 58 M-irreducible p. 103 inner measure (integral); p. 8 , p. 15 y * v convolution of measures; p. 68 y ® v product of measures; p. 66 non-atomic p. 59 o composition (of functions) p paving of positive (cozero) sets; P- 50 p p. 107 P(x, i4) p. 102 P”(a?, A) p. 102 Pr fidi; p. 90 setwise fidi; p. 93 *4 A--. 1 n V'P image measure of P under the map 'F ; p. 64 cp-irreducible p. 102 »Iü restriction of the function ¥ to the set A p-smooth p. 64 paving a non-empty class of subsets; p. 7 point process p. 89 product measure p. 66 product uniformity p. 63 pseudometric p. 61 Q. p. 89 $ the space of rational numbers R the real line * extended real line Rn Euclidean n-space i?-invariantP-invariant function p.
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