Theory of Charges
A Study of Finitely Additive Measures
Published on 10. May 1983
315 pages
978-0-08-087428-9 (ISBN)
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Theory of Charges
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09/1997
Academic Press
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Content
- Front Cover
- Theory of Charges: A Study of Finitely Additive Measures
- Copyright Page
- Contents
- Foreword
- Preface
- CHAPTER 1 PRELIMINARIES
- 1.1 Classes of sets
- 1.2 Set theoretical concepts
- 1.3 Topological concepts
- 1.4 Boolean algebras
- 1.5 Functional analytic concepts
- CHAPTER 2 CHARGES
- 2.1 Basic concepts
- 2.2 The space of all bounded charges, ba(O,F)
- 2.3 Measures
- 2.4 The space of all bounded measures, ca(O,F)
- 2.5 Jordan Decomposition theorem
- 2.6 Hahn Decomposition theorem
- CHAPTER 3 EXTENSIONS OF CHARGES
- 3.1 Real valued set functions and induced functionals
- 3.2 Real partial charges and their extensions
- 3.3 Extension procedure of Los and Marczewski
- 3.4 Extension of partial charges in the general case
- 3.5 Miscellaneous extensions
- 3.6 Common extensions
- CHAPTER 4 INTEGRATION
- 4.1 Total variation and outer charges
- 4.2 Null sets and null functions
- 4.3 Hazy convergence
- 4.4 D-integral
- 4.5 S-integral
- 4.6 Lp- spaces
- 4.7 ba(O,F) as a dual space
- CHAPTER 5 NONATOMIC CHARGES
- 5.1 Basic concepts
- 5.2 Sobczyk-Hammer Decomposition theorem
- 5.3 Existence of nonatomic charges
- 5.4 Denseness
- CHAPTER 6 ABSOLUTE CONTINUITY
- 6.1 Absolute continuity and singularity
- 6.2 Lebesgue Decomposition theorem
- 6.3 Radon-Nikodym theorem
- CHAPTER 7 Vp-SPACES
- 7.1 Lp- spaces-An overview
- 7.2 Vp- spaces
- 7.3 Duals of Vp- spaces
- 7.4 Strong Convergence
- 7.5 Weak Convergence
- CHAPTER 8 NIKODYM THEOREM, WEAK CONVERGENCE AND VITALI-HAHN-SAKS THEOREM
- 8.1 Nikodym and Vitali-Hahn-Saks theorems in the classical case
- 8.2 Examples
- 8.3 Phillips' lemma
- 8.4 Nikodym theorem
- 8.5 Norm bounded sets in the presence of uniform absolute continuity
- 8.6 A decomposition theorem
- 8.7 Weak convergence
- 8.8 Vitali-Hahn-Saks theorem
- CHAPTER 9 THE DUAL OF ba (O, F) AND THE REFINEMENT INTEGRAL
- 9.1 Refinement integral
- 9.2 The dual of ba(O,F)
- CHAPTER 10 PURE CHARGES
- 10.1 Definitions and properties
- 10.2 A decomposition theorem
- 10.3 Pure charges on s-fields
- 10.4 Examples
- 10.5 Pure charges on Boolean algebras
- CHAPTER 11 RANGES OF CHARGES
- 11.1 Ranges of bounded charges on fields
- 11.2 Ranges of charges on s-fields
- 11.3 Cardinalities of ranges of charges
- 11.4 Charges with closed range
- 11.5 Charges whose ranges are neither Lebesgue measurable nor have the property of Baire
- CHAPTER 12 ON LIFTING
- Appendix 1 Notes and Comments
- Appendix 2 Selected Annotated Bibliography
- Appendix 3 Some Set Theoretic Nomenclature
- Index of Symbols and Function Spaces
- Subject Index
