Vector and Operator Valued Measures and Applications (eBook)

Hugh B. Maynard, Don H. Tucker (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
474 Seiten
Elsevier Science (Verlag)
978-1-4832-6102-7 (ISBN)

Infos

Vector and Operator Valued Measures and Applications is a collection of papers presented at the Symposium on Vector and Operator Valued Measures and Applications held in Alta, Utah, on August 7-12, 1972. The symposium provided a forum for discussing vector and operator valued measures and their applications to various areas such as stochastic integration, electrical engineering, control theory, and scattering theory. Comprised of 37 chapters, this volume begins by presenting two remarks related to the result due to Kolmogorov: the first is a theorem holding for nonnegative definite functions from T X T to C (where T is an arbitrary index set), and the second applies to separable Hausdorff spaces T, continuous nonnegative definite functions ? from T X T to C, and separable Hilbert spaces H. The reader is then introduced to the extremal structure of the range of a controlled vector measure ? with values in a Hausdorff locally convex space X over the field of reals; how the theory of vector measures is connected with the theory of compact and weakly compact mappings on certain function spaces; and Daniell and Daniell-Bochner type integrals. Subsequent chapters focus on the disintegration of measures and lifting; products of spectral measures; and mean convergence of martingales of Pettis integrable functions. This book should be of considerable use to workers in the field of mathematics.

Inhalt

Front Cover 1
Vector and Operator Valued Measures and Applications 4
Copyright Page 5
Table of Contents 6
PARTCIPANTS 10
PREFACE 16
CHAPTER 1. SOME REMARKS ON A KOLMOGOV 18
1 troduction 18
2. The First Remarks 18
3. The Second Remark 20
References 21
CHAPTER 2. ON EXPOSED POINTS OF THE RANGE 24
0 Introduction 24
1 Terminology and Notation 25
2. Extreme Points 27
Exposets Points 32
References 38
CHAPTER 3. A SURVEY OF SOME RECENT RESULTS ON COMPACT MAPPINGS 40
0 Introduction 40
1 The Space C(S,E) 40
2 The Space B(O,S,.) 43
3 Extensions to Nonlinear Mappings 44
Reference 48
CHAPTER 4. A SUVEY OF DANIELL INTEGRATION 50
1 Extension Theory 50
2 Generalizations 52
3 Examples 53
4 Measurability 55
References 58
CHAPTER 5. DANIELL AND DANIELL-BOCHNER TYPE INTEGRALS 60
1 Daniell Integrals 60
2 Daniell-Bochner Type Integrals 64
References 66
CHAPTER 6. EQUICONTINUITY, ABSOLUTE CONTINUITY AND WEAK COMACTNESS IN MEASURE THEORY 68
0 Introduction 68
1 Definitions and Notation 68
2 The Main Results 69
References 77
CHAPTER 7. INDEPENDENCE AND CONTRACTIVE PROJECTIONS ON L . 80
0 Introduction 80
1 Contractive Projections with Contractive Conplements 81
2 Commutativity 82
References 84
CHAPTER 8. DISINTEGRATION OF MEASURES AND LIFTING 86
0. Introduction 86
1 Lifting 87
2 Applications of Lifting 93
3 Disintegration of Measures 94
References 99
CHAPTER 9. THE RADON-NIKODYM THEOREM FOR VECTOR MEASURES WITH VALUES IN THE DUALS OF S(M NUCLEAR SPACES 102
Abstract 102
0 Introduction 102
1 Preliminaries 103
2 The Radon-Nikodym Theorem 105
3 Application 110
References 111
CHAPTER 10. GROTHENDIBCK SPACES AND VECTOR MEASURES 114
1 Introduction 114
2 Main Results 115
3 Problems 121
Appendix 123
References 124
CHAPTER 11. LINEAR OPERATIONS ON Lp -SPACES 126
1 Introduction 126
2 Successive Extensions of Theorem 1 127
3 Vector Measures. Variation and Semi-variation 130
4 Integration with Respect to Vector Measures 132
5 Linear Operations on LpE and M8E 133
6 A General Radon-Nikodym Theorem 135
7 Integral Representation of Linear Operations on LpE , 1 = p < 8
References 138
CHAPTER 12. A NOTE ON PRODUCTS OF SPECTRAL MEASURES 142
References 143
CHAPTER 13. VECTOR VALUED MEASURES AND DECOMPOSING MAPS 144
0 Introduction 144
1 Vector Valued Measures 144
2 Two Lemmas for the Real Case 145
3 Linear Random Processes 145
4 A Radon-Nikodym Theorem 147
References 149
CHAPTER 14. THE YOSIDA-HEWIIT DECOMPOSITION AS AN ERGODIC THEOREM 150
References 156
CHAPTER 15. STOCHASTIC INTEGRATION 158
1 Wiener Integral 158
2 Stochastic Integral 159
3 Stochastic Differentials 161
4 Multiple Wiener Integral 162
References 164
CHAPTER 16. THE SPECTRAL INTEGRAL IN SCATTERING THEORY 166
References 170
CHAPTER 17. PRE-MEASURES ON LATTICES OF SETS - II 172
References 181
CHAPTER 18. ON THE BOCHNER INTEGRAL 182
0 Introduction 182
1 The Bochner Integral 182
2 Bochner Indefinite Integrals 187
CHAPTER 19. THE EXTENSION AND CLOSURE OF VECTOR MEASURE 192
References 203
CHAPTER 20. HIGHER-ORDER DIFFERENTIABILITY OF THE NORM IN Lp(E) 208
Abstract 208
1 introduction 208
2 Definitions and Notation 208
3 Summary of Results 210
4 Exanple 210
References 211
CHAPTER 21. PERMANENCE PROPERTIES OF ABSOLUTE CONTINUITY CONDITIONS 214
References 222
CHAPTER 22. ON THE STOJCTURE OF L2,. 224
1 Introduction 224
2 Preliminaries 226
3 The Space L2M and its Conpleteness 227
4 Hilbert ian Structure of L2M 230
References 231
CHAPTER 23. RADON-NIKODYM THEORY FOR OPERATOR VALUED MEASURES AND A THEOREM OF FOURES & SEGAL
1 Nature of the Problem and Scope of This Report 234
2 Resume of Quasi-isometric Measures 237
Radon-Nikodym Theory at Level . 239
4 Operational Calculus for the Class 2TC 242
5 Radon-Nikodym Theory at Level A Hypermaximality
6 The Levels C and D 246
References 249
CHAPTER 24. A GENERAL RADON-NIKODYM THEOREM 250
Applications to Bochner and Strongly Measurable Pettis Integrable Functions 260
References 263
CHAPTER 25. STOCHASTIC INTEGRATION 264
1 Motivation 264
2 Definition of the Integral 266
3 Elementary Theorems 271
4 Existence Theorem for the First-Order Integral 274
5 Conparison with Other Definitions 280
6 Existence of Higher-Order Integrals 282
7 The I t o Differentiation Formula 289
8 Differential Equations 292
9 The Canonical Form 295
References 297
CHAPTER 26. STOCHASTIC INTEGRAL AND VECTOR VALUED MEASURES 300
0 Introduction 300
1 Notations - Definitions 301
2 Stochastic Measures 302
3 Stochastic Measures Generated by a Stochastic Process 303
4 Existence of the Generated Stochastic Measure 307
5 Local Stochastic Measure Generated by a Local Martingale 310
6 Definition of the Stochastic Integral 311
References 312
CHAPTER 27. VECTOR-VALUED MEASURES RELATED TO A GENERALIZED CONTINUOUS HOMOGENEOUS RANDOM FIELD 314
References 318
CHAPTER 28. RH4ARKS ON A RADON-NIKODYM THEORY FOR VECTOR MEASURES 320
1 Introduction 320
2 Radon-Nikodym Theorems for Vector Measures 321
3 A Special Class of Vector Measures 327
4 Application 330
5 Remarks and Problems 332
References 332
CHAPTER 29. CAUSAL FACTORIZATION, SHIFT OPERATORS AND THE SPECTRAL MULTIPLICITY FUNCTION 336
Abstract 336
1 Introduction 336
2 Uniform Resolution Space 340
3 Causal Factorization 346
4 Conclusions 351
References 351
CHAPTER 30. INTEGRAL AS A CERTAIN TYPE OF A POSITIVE DEFINITE FUNCTION 354
Abstract 354
References 354
CHAPTER 31. VECTOR MEASURES AND THE SPECTRAL THEOREM 356
Abstract 356
References 358
CHAPTER 32. L-ORTHOGONALLY SCATTERED MEASURES 360
1 360
2 362
3 363
4 364
5 365
6 366
References 367
CHAPTER 33. ABSOLUTELY SIMIING OPERATORS ON CX(S) 368
References 370
CHAPTER 34. S-BOUNDED ADDITIVE SET FUNCTIONS 372
Abstract 372
0 Introduction 372
1 Control Measures 373
2 Uniform Absolute Continuity 376
3 Boundedness 378
4 Appendix 381
References 381
CHAPTER 35. SOME REMARKS ON THE NOTION OF "DERIVATIVES 384
1 Introduction 384
2 A Survey of the Problem 385
3 Multifunctions 392
References 394
CHAPTER 36. A SURVEY OF MEAN CONVERGENCE OF MARTINGALES OF PETTIS INTEGRABLE FUNCTIONS 396
0 Introduction 396
1 Convergence Within the Completion of .1(µ,-.) 397
2 Convergence to a Function in .1(µ,X) 397
References 401
CHAPTER ON THE CONSTRUCTION OF THE HILBERT SPACE L2,M FOR AN OPERATOR-VALUED MEASURE M 404
0 Introduction 404
1 Construction of the Space L2,M 406
2 Approximation of Functions in L2,M by Sinple Functions 410
References 412
BIBLIOGRAPHY 416

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 1-4832-6102-6 / 1483261026
• ISBN-13: 978-1-4832-6102-7 / 9781483261027