CONDITIONAL MEASURES AND APPLICATIONS
CONTENTS CHAPTER 1: THE CONCEPT OF CONDITIONING 1 1 INTRODUCTION 1 2 CONDITIONAL PROBABILITY GIVEN A PARTITION 1 3 CONDITIONAL EXPECTATION: ELEMENTARY CASE 1 4 CONDITIONING WITH DENSITIES 1 5 CONDITIONAL PROBABILITY SPACES: ?RST STEPS 1 6 BIBLIOGRAPHICAL NOTES CHAPTER 2: THE KOLMOGOROV FORMULATION AND ITS PROPERTIES 2 1 INTRODUCTION OF THE GENERAL CONCEPT 2 2 BASIC PROPERTIES OF CONDITIONAL EXPECTATIONS 2 3 CONDITIONAL PROBABILITIES IN THE GENERAL CASE 2 4 REMARKS ON THE INCLUSION OF PREVIOUS CONCEPTS 2 5 CONDITIONAL INDEPENDENCE AND RELATED CONCEPTS 2 6 BIBLIOGRAPHICAL NOTES CHAPTER 3: COMPUTATIONAL PROBLEMS ASSOCIATED WITH CONDITIONING 3 1 INTRODUCTION 3 2 SOME EXAMPLES WITH MULTIPLE SOLUTIONS: PARADOXES 3 3 DISSECTION OF PARADOXES 3 4 SOME METHODS OF COMPUTATION 3 5 REMARKS ON TRADITIONAL CALCULATIONS OF CONDITIONAL MEASURES 3 6 BIBLIOGRAPHICAL NOTES CHAPTER 4: AN AXIOMATIC APPROACH TO CONDITIONAL PROBABILITY 4 1 INTRODUCTION 4 2 AXIOMATIZATION OF CONDITIONING BASED ON PARTITIONS 4 3 STRUCTURE OF THE NEW CONDITIONAL PROBABILITY FUNCTIONS 4 4 SOME APPLICATIONS 4 5 DI?CULTIES WITH EARLIER EXAMPLES PERSIST 4 6 BIBLIOGRAPHICAL NOTES CHAPTER 5: REGULARITY OF CONDITIONAL MEASURES 5 1 INTRODUCTION 5 2 EXISTENCE OF REGULAR CONDITIONAL PROBABILITIES: GENERALITIES 5 3 SPECIAL SPACES ADMITTING REGULAR CONDITIONAL PROBABILITIES 5 4 DISINTEGRATION OF PROBABILITY MEASURES AND REGULAR CONDITIONING 5 5 FURTHER RESULTS ON DISINTEGRATION 5 6 EVALUATION OF CONDITIONAL EXPECTATIONS BY FOURIER ANALYSIS 5 7 FURTHER EVALUATIONS OF CONDITIONAL EXPECTATIONS 5 8 BIBLIOGRAPHICAL NOTES CHAPTER 6: SU?CIENCY 6 1 INTRODUCTION 6 2 CONDITIONING RELATIVE TO FAMILIES OF MEASURES 6 3 SU?CIENCY: THE DOMINATED CASE 6 4 SU?CIENCY: THE UNDOMINATED CASE 6 5 SU?CIENCY: ANOTHER APPROACH TO THE UNDOMINATED CASE 6 6 BIBLIOGRAPHICAL NOTES CHAPTER 7: ABSTRACTION OF KOLMOGOROV’S FORMULATION 7 1 INTRODUCTION 7 2 INTEGRATION RELATIVE TO CONDITIONAL MEASURES AND FUNCTION SPACES 7 3 FUNCTIONAL CHARACTERIZATIONS OF CONDITIONING 7 4 INTEGRAL REPRESENTATIONS OF CONDITIONAL EXPECTATIONS 7 5 R´ENYI’S FORMULATION AS A SPECIALIZATION OF THE ABSTRACT VERSION 7 6 CONDITIONAL MEASURES AND DI?ERENTIATION 7 7 BIBLIOGRAPHICAL NOTES CHAPTER 8: PRODUCTS OF CONDITIONAL MEASURES 8 1 INTRODUCTION 8 2 A GENERAL FORMULATION OF PRODUCTS 8 3 GENERAL PROJECTIVE LIMIT THEOREMS 8 4 SOME CONSEQUENCES 8 5 REMARKS ON CONDITIONING, DISINTEGRATION, AND LIFTING 8 6 BIBLIOGRAPHICAL NOTES CHAPTER 9: APPLICATIONS TO MARTINGALES AND MARKOV PROCESSES 9 1 INTRODUCTION 9 2 SET MARTINGALES 9 3 MARTINGALE CONVERGENCE 9 4 MARKOV PROCESSES: SOME BASIC RESULTS 9 5 FURTHER PROPERTIES OF MARKOV PROCESSES 9 6 BIBLIOGRAPHICAL NOTES CHAPTER 10: APPLICATIONS TO MODERN ANALYSIS 10 1 INTRODUCTION AND MOTIVATION 10 2 CONDITIONAL MEASURES AND POTENTIAL KERNELS 10 3 REYNOLDS OPERATORS AND CONDITIONAL EXPECTATIONS 10 4 BISTOCHASTIC OPERATORS AND CONDITIONING 10 5 CONTRACTIVE PROJECTIONS AND CONDITIONAL EXPECTATIONS 10 6 BIBLIOGRAPHICAL NOTES CHAPTER 11: CONDITIONING IN GENERAL STRUCTURES 11 1 INTRODUCTION 11 2 AVERAGINGS IN CONES OF POSITIVE FUNCTIONS 11 3 AVERAGING OPERATORS ON FUNCTION ALGEBRAS 11 4 CONDITIONING IN OPERATOR ALGEBRAS 11 5 FREE INDEPENDENCE AND A BIJECTION IN OPERATOR ALGEBRAS 11 6 SOME APPLICATIONS OF NONCOMMUTATIVE CONDITIONING 11 7 BIBLIOGRAPHICAL NOTES REFERENCES NOTATIONS