Jacobs / Birnbaum / Lukacs Measure and Integral

1. Auflage 2014
ISBN: 978-1-4832-6304-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 592 Seiten, Web PDF

Probability and Mathematical Statistics: Measure and Integral provides information pertinent to the general mathematical notions and notations. This book discusses how the machinery of ?-extension works and how ?-content is derived from ?-measure. Organized into 16 chapters, this book begins with an overview of the classical Hahn-Banach theorem and introduces the Banach limits in the form of a major exercise. This text then presents the Daniell extension theory for positive ?-measures. Other chapters consider the transform of ?-contents and ?-measures by measurable mappings and kernels. This text is also devoted to a thorough study of the vector lattice of signed contents. This book discusses as well an abstract regularity theory and applied to the standard cases of compact, locally compact, and Polish spaces. The final chapter deals with the rudiments of the Krein-Milman theorem, along with some of their applications. This book is a valuable resource for graduate students.

Autoren/Hrsg.

Jacobs, Konrad

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Measure and Integral;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;10
6;CHAPTER 0. BASIC NOTIONS AND NOTATION;18
6.1;1. SETS, RELATIONS, AND MAPPINGS;18
6.2;2. VECTOR SPACES AND VECTOR LATTICES. BANACH SPACES;29
6.3;3. TOPOLOGICAL SPACES;36
7;CHAPTER I. POSITIVE CONTENTS AND MEASURES;43
7.1;1. CONTENTS ON SET RINGS;43
7.2;2. s-CONTENTS;51
7.3;3. EUDOXOS EXTENSION OF CONTENTS;55
7.4;4. s-RINGS, LOCAL s-RINGS, AND s-FIELDS;59
7.5;5. UNIQUENESS OF EXTENSION OF s-CONTENTS;66
7.6;6. THE HAHN-BANACH THEOREM;68
7.7;7. ELEMENTARY DOMAINS;71
7.8;8. MEASURES ON ELEMENTARY DOMAINS;75
7.9;9. RIEMANN (EUDOXOS) EXTENSION OF POSITIVE MEASURES;80
7.10;10. MEASURE SPACES OVER TOPOLOGICAL SPACES;84
7.11;11. THE EXTENSION PROBLEM FOR MEASURE SPACES;87
8;CHAPTER II. EXTENSION OF s-CONTENTS AFTER CARATHÉODORY;89
8.1;1. THE OUTER CONTENT DERIVED FROM A GIVEN CONTENT;89
8.2;2. ADDITIVE DECOMPOSERS;93
8.3;3. ROUTINE EXTENSION FROM LOCAL s-RINGS TO s-RINGS;97
8.4;4. MINIMAL EXTENSION TO A s-FIELD;98
8.5;5. DEFINITION OF A s-CONTENT FROM LOCAL DATA;99
8.6;6. COMPLETION;101
9;CHAPTER III. EXTENSION OF POSITIVE s- AND t-MEASURES, AFTER DANIELL;105
9.1;1. EXTENSION STEP I: s-UPPER AND s-LOWER FUNCTIONS, MONOTONE EXTENSION;105
9.2;2. EXTENSION STEP II: SQUEEZING-IN AND THE DEFINITION OF INTEGRABLE FUNCTIONS;112
9.3;3. THE UPPER AND THE LOWER INTEGRAL;115
9.4;4. NULLFUNCTIONS AND NULLSETS;119
9.5;5. BASIC THEOREMS FOR THE INTEGRAL;124
9.6;6. THE s-CONTENT DERIVED FROM A s-MEASURE;134
9.7;7. t-EXTENSION OF A t-MEASURE;140
9.8;8. MEASURABILITY OF REAL AND COMPLEX FUNCTIONS;142
9.9;9. MEASURABILITY AND INTEGRABILITY;149
9.10;10. INTEGRATION OF COMPLEX-VALUED FUNCTIONS;151
9.11;11. THE REAL AND THE COMPLEX HILBERT SPACE L2;153
9.12;12. STOCHASTIC CONVERGENCE AND UNIFORM INTEGRABILITY;157
10;CHAPTER IV. TRANSFORM OF s-CONTENTS;165
10.1;1. MEASURABLE MAPPINGS;167
10.2;2. TRANSFORM OF s-ADDITIVE FUNCTIONS;171
10.3;3. ERGODIC THEOREMS;175
10.4;4. KERNELS;198
11;CHAPTER V.
CONTENTS AND MEASURES IN TOPOLOGICAL SPACES. PART I: REGULARITY;213
11.1;1. THE GENERAL CONCEPT OF REGULARITY;213
11.2;2. REGULARITY OF s-CONTENTS IN TOPOLOGICAL SPACES;217
11.3;3. REGULARITY OF s-CONTENTS IN COMPACT SPACES;218
11.4;4. REGULARITY OF s-CONTENTS IN LOCALLY COMPACT SPACES;222
11.5;5. REGULARITY IN POLISH SPACES;226
12;CHAPTER VI. CONTENTS AND MEASURES IN PRODUCT SPACES;229
12.1;1. SET SYSTEMS IN PRODUCT SPACES;230
12.2;2. TWO FACTORS;236
12.3;3. FINITELY MANY FACTORS;243
12.4;4. COUNTABLY MANY FACTORS;246
12.5;5. ARBITRARILY MANY FACTORS;250
12.6;6. INDEPENDENCE;259
12.7;7. MARKOVIAN SEMIGROUPS AND THEIR PATH STRUCTURE;263
13;CHAPTER VII. SET FUNCTIONS IN GENERAL;282
13.1;1. BASIC NOTIONS FOR SET FUNCTIONS;283
13.2;2. THE ADDITIVE AND s-ADDITIVE PARTS OF A SUPERADDITIVE
SET FUNCTION;287
13.3;3. s-ADDITIVITY. THE VITALI–HAHN–SAKS THEOREM;290
13.4;4. TOTAL VARIATION;296
14;CHAPTER VIII. THE VECTOR LATTICE OF SIGNED CONTENTS;299
14.1;1. SIGNED CONTENTS AND SIGNED s-CONTENTS;300
14.2;2. HAHN DECOMPOSITIONS;307
14.3;3. ABSOLUTE CONTINUITY OF SIGNED s-CONTENTS;309
14.4;4. LEBESGUE DECOMPOSITIONS;313
14.5;5. THE RADON-NIKODYM THEOREM FOR SIGNED s-CONTENTS WITH FINITE TOTAL VARIATION;316
14.6;6. CONDITIONAL EXPECTATIONS;321
14.7;7. MARTINGALES, SUBMARTINGALES, AND SUPERMARTINGALES;327
14.8;8. THE RADON-NIKODYM PROBLEM;335
15;CHAPTER IX. THE VECTOR LATTICE OF SIGNED MEASURES;342
15.1;1. ABSTRACT VECTOR LATTICES AND THEIR DUALS;342
15.2;2. SIGNED MEASURES;349
15.3;3. ABSOLUTE CONTINUITY OF MEASURES;354
15.4;4. SIGNED CONTENTS AND SIGNED MEASURES;357
16;CHAPTER X.
THE SPACES Lp;361
16.1;1. THE SPACES Lpm (1. p . p 8);362
16.2;2. DUALITY OF THE SPACES Lmp (1. p . 8);367
17;CHAPTER XI. CONTENTS AND MEASURES IN TOPOLOGICAL SPACES. PART II: THE WEAK TOPOLOGY;374
17.1;1. THE WEAK TOPOLOGY FOR s-CONTENTS IN ARBITRARY TOPOLOGICAL SPACES;375
17.2;2. THE WEAK TOPOLOGY FOR s-CONTENTS IN POLISH SPACES;383
18;CHAPTER XII. THE HAAR MEASURE ON LOCALLY COMPACT
GROUPS;391
18.1;1. THE HAAR MEASURE ON COMPACT GROUPS;392
18.2;2. DEFINITION AND BASIC MACHINERY OF LOCALLY COMPACT GROUPS;400
18.3;3. THE HAAR MEASURE ON LOCALLY COMPACT GROUPS;402
19;CHAPTER XIII. SOUSLIN SETS, ANALYTIC SETS, AND CAPACITIES;419
19.1;1. SOUSLIN EXTENSIONS;420
19.2;2. SOUSLIN SETS AND ANALYTIC SETS IN POLISH SPACES;428
19.3;3. CAPACITIES;438
19.4;4. THE CAPACITY APPROACH TO s-CONTENT EXTENSION;447
19.5;5. THE MEASURABLE CHOICE THEOREM;449
20;CHAPTER XIV.
ATOMS, CONDITIONAL ATOMS, AND ENTROPY;453
20.1;1. ATOMS AND CONDITIONAL ATOMS;453
20.2;2. ENTROPY;464
21;CHAPTER XV. CONVEX COMPACT SETS AND THEIR EXTREMAL POINTS;474
21.1;1. LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES;475
21.2;2. BARYCENTERS;483
21.3;3. APPLICATIONS;495
22;CHAPTER XVI.
LIFTING;501
22.1;1. THE NOTION AND EXISTENCE OF A LIFTING;502
22.2;2. STRONG LIFTING;512
22.3;3. APPLICATIONS;516
23;APPENDIX A: THE PERRON-WARD INTEGRAL AND RELATED CONCEPTS;532
23.1;1. NOTATIONS;533
23.2;2. THE S-INTEGRAL;534
23.3;3. THE V-INTEGRAL;536
23.4;4. THE PERRON-WARD INTEGRAL;538
23.5;5. MONOTONE CONVERGENCE;541
23.6;6. RELATION TO THE DANIELL INTEGRAL;543
23.7;7. SOME RESULTS ON THE S-INTEGRAL;548
24;APPENDIX B:
CONTENTS WITH GIVEN MARGINALS;551
24.1;1. THE FORD-FULKERSON THEOREM;551
24.2;2. MATRICES WITH GIVEN MARGINALS;555
24.3;3. CONTENTS AND s-CONTE NTS WITH GIVEN MARGINALS;557
24.4;4. RESULTS INVOLVING TOPOLOGY;562
25;SELECTED BIBLIOGRAPHY;566
26;INDEX;578
27;Probability and Mathematical Statistics;593