Theory of Charges (eBook)

A Study of Finitely Additive Measures

K.P.S. Bhaskara Rao, M. Bhaskara Rao (Autoren)

eBook Download: PDF

1983 | 1. Auflage
315 Seiten
Elsevier Science (Verlag)
978-0-08-087428-9 (ISBN)

Infos

Theory of Charges

Inhalt

Front Cover 1
Theory of Charges: A Study of Finitely Additive Measures 4
Copyright Page 5
Contents 10
Foreword 6
Preface 8
CHAPTER 1 PRELIMINARIES 12
1.1 Classes of sets 12
1.2 Set theoretical concepts 24
1.3 Topological concepts 26
1.4 Boolean algebras 29
1.5 Functional analytic concepts 34
CHAPTER 2 CHARGES 46
2.1 Basic concepts 46
2.2 The space of all bounded charges, ba(O,F) 54
2.3 Measures 58
2.4 The space of all bounded measures, ca(O,F) 61
2.5 Jordan Decomposition theorem 63
2.6 Hahn Decomposition theorem 67
CHAPTER 3 EXTENSIONS OF CHARGES 69
3.1 Real valued set functions and induced functionals 69
3.2 Real partial charges and their extensions 75
3.3 Extension procedure of Los and Marczewski 81
3.4 Extension of partial charges in the general case 87
3.5 Miscellaneous extensions 89
3.6 Common extensions 93
CHAPTER 4 INTEGRATION 96
4.1 Total variation and outer charges 96
4.2 Null sets and null functions 98
4.3 Hazy convergence 103
4.4 D-integral 107
4.5 S-integral 126
4.6 Lp- spaces 132
4.7 ba(O,F) as a dual space 144
CHAPTER 5 NONATOMIC CHARGES 152
5.1 Basic concepts 152
5.2 Sobczyk-Hammer Decomposition theorem 155
5.3 Existence of nonatomic charges 161
5.4 Denseness 167
CHAPTER 6 ABSOLUTE CONTINUITY 170
6.1 Absolute continuity and singularity 170
6.2 Lebesgue Decomposition theorem 177
6.3 Radon-Nikodym theorem 180
CHAPTER 7 Vp-SPACES 189
7.1 Lp- spaces—An overview 189
7.2 Vp- spaces 196
7.3 Duals of Vp- spaces 204
7.4 Strong Convergence 208
7.5 Weak Convergence 211
CHAPTER 8 NIKODYM THEOREM, WEAK CONVERGENCE AND VITALI–HAHN–SAKS THEOREM 214
8.1 Nikodym and Vitali–Hahn–Saks theorems in the classical case 215
8.2 Examples 216
8.3 Phillips' lemma 217
8.4 Nikodym theorem 220
8.5 Norm bounded sets in the presence of uniform absolute continuity 224
8.6 A decomposition theorem 227
8.7 Weak convergence 229
8.8 Vitali–Hahn–Saks theorem 237
CHAPTER 9 THE DUAL OF ba (O, F) AND THE REFINEMENT INTEGRAL 242
9.1 Refinement integral 242
9.2 The dual of ba(O,F) 245
CHAPTER 10 PURE CHARGES 251
10.1 Definitions and properties 251
10.2 A decomposition theorem 252
10.3 Pure charges on s-fields 254
10.4 Examples 255
10.5 Pure charges on Boolean algebras 257
CHAPTER 11 RANGES OF CHARGES 260
11.1 Ranges of bounded charges on fields 260
11.2 Ranges of charges on s-fields 263
11.3 Cardinalities of ranges of charges 267
11.4 Charges with closed range 268
11.5 Charges whose ranges are neither Lebesgue measurable nor have the property of Baire 275
CHAPTER 12 ON LIFTING 279
Appendix 1 Notes and Comments 283
Appendix 2 Selected Annotated Bibliography 293
Appendix 3 Some Set Theoretic Nomenclature 316
Index of Symbols and Function Spaces 317
Subject Index 320

Produkt-Details

• Erscheint lt. Verlag: 10.5.1983
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geophysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• ISBN-10: 0-08-087428-2 / 0080874282
• ISBN-13: 978-0-08-087428-9 / 9780080874289