Theory of Charges: A Study of Finitely Additive Measures

Theory of Charges: A Study of Finitely Additive Measures......Page 4
Copyright Page......Page 5
Contents......Page 10
Foreword......Page 6
Preface......Page 8
1.1 Classes of sets......Page 12
1.2 Set theoretical concepts......Page 24
1.3 Topological concepts......Page 26
1.4 Boolean algebras......Page 29
1.5 Functional analytic concepts......Page 34
2.1 Basic concepts......Page 46
2.2 The space of all bounded charges, ba(Ω,F)......Page 54
2.3 Measures......Page 58
2.4 The space of all bounded measures, ca(Ω,F)......Page 61
2.5 Jordan Decomposition theorem......Page 63
2.6 Hahn Decomposition theorem......Page 67
3.1 Real valued set functions and induced functionals......Page 69
3.2 Real partial charges and their extensions......Page 75
3.3 Extension procedure of Los and Marczewski......Page 81
3.4 Extension of partial charges in the general case......Page 87
3.5 Miscellaneous extensions......Page 89
3.6 Common extensions......Page 93
4.1 Total variation and outer charges......Page 96
4.2 Null sets and null functions......Page 98
4.3 Hazy convergence......Page 103
4.4 D-integral......Page 107
4.5 S-integral......Page 126
4.6 Lp- spaces......Page 132
4.7 ba(Ω,F) as a dual space......Page 144
5.1 Basic concepts......Page 152
5.2 Sobczyk-Hammer Decomposition theorem......Page 155
5.3 Existence of nonatomic charges......Page 161
5.4 Denseness......Page 167
6.1 Absolute continuity and singularity......Page 170
6.2 Lebesgue Decomposition theorem......Page 177
6.3 Radon-Nikodym theorem......Page 180
7.1 Lp- spaces—An overview......Page 189
7.2 Vp- spaces......Page 196
7.3 Duals of Vp- spaces......Page 204
7.4 Strong Convergence......Page 208
7.5 Weak Convergence......Page 211
CHAPTER 8 NIKODYM THEOREM, WEAK CONVERGENCE AND VITALI–HAHN–SAKS THEOREM......Page 214
8.1 Nikodym and Vitali–Hahn–Saks theorems in the classical case......Page 215
8.2 Examples......Page 216
8.3 Phillips' lemma......Page 217
8.4 Nikodym theorem......Page 220
8.5 Norm bounded sets in the presence of uniform absolute continuity......Page 224
8.6 A decomposition theorem......Page 227
8.7 Weak convergence......Page 229
8.8 Vitali–Hahn–Saks theorem......Page 237
9.1 Refinement integral......Page 242
9.2 The dual of ba(Ω,F)......Page 245
10.1 Definitions and properties......Page 251
10.2 A decomposition theorem......Page 252
10.3 Pure charges on σ-fields......Page 254
10.4 Examples......Page 255
10.5 Pure charges on Boolean algebras......Page 257
11.1 Ranges of bounded charges on fields......Page 260
11.2 Ranges of charges on σ-fields......Page 263
11.3 Cardinalities of ranges of charges......Page 267
11.4 Charges with closed range......Page 268
11.5 Charges whose ranges are neither Lebesgue measurable nor have the property of Baire......Page 275
CHAPTER 12 ON LIFTING......Page 279
Appendix 1 Notes and Comments......Page 283
Appendix 2 Selected Annotated Bibliography......Page 293
Appendix 3 Some Set Theoretic Nomenclature......Page 316
Index of Symbols and Function Spaces......Page 317
Subject Index......Page 320