Measure and Integration Theory on Infinite-Dimensional Spaces (eBook)

Front Cover 1
Measure and Integration Theory on Infinite-Dimensional Spaces 4
Copyright Page 5
Contents 6
Foreword 8
Preface 10
Chapter I. Some Supplementary Background in Measure Theory 12
§1.1. Some Measure-Theoretic Concepts 13
§1.2. Localizable Measure Spaces 35
§1.3. The Kolmogorov Theorem 41
§1.4. Kakutani Distance 52
Chapter II. Representation of Positive Functionals and Operator Rings 60
§2.1. Topological Algebras with Involution: Fundamental Concepts 61
§2.2. Representation of Positive Functionals on Seminormed Algebras 73
§2.3. Weakly Closed Operator Algebras: Fundamental Concepts 82
§2.4. Representation of Commutative Weakly Closed Operator Rings 90
Chapter III. Harmonic Analysis on Groups with Quasi-Invariant Measures 114
§3.1. Basic Properties of Quasi-Invariant Measures 116
§3.2. Characters and Quasi-Characters 146
§3.3. Integral Representation of Positive Definite Functions on Groups 175
§3.4. L2-Fourier Transforms 196
Chapter IV. Quasi-Invariant Measures and Harmonic Analysis on Linear Topological Spaces 224
§4.1. Quasi-Invariant Measures on Linear Topological Spaces 225
§4.2. Linear and Quasi-Linear Functionals on Linear Spaces 243
§4.3. Continuous Positive Definite Functions on Linear Topological Spaces 265
Chapter V. Gaussian Measures 291
§5.1. Some Properties of Gaussian Measures 292
§5.2. Equivalence and Perpendicularity of Gaussian Measures 306
55.3. Gaussian Measures on Linear Spaces 322
§5.4. Fourier-Gauss Transforms 337
Chapter VI. Representation of Commutation Relations in Bose-Einstein Fields 346
§6.1. Representations of the Commutation Relations in Quantum Mechanics 347
§6.2. Quasi-Invariant Measures Applied to Representations of the Commutation Relations in Bose-Einstein Fields 363
§6.3. The Relation of Gaussian Measures and Rotationally Invariant Measures to Conventional Free-Field Systems 380
Appendix I. Background Material on Topological Groups and Linear Topological Spaces 392
§1.1. Pseudometrics, Convex Functions, and Pseudonorms 392
§1.2. Some Properties of Semicontinuous Functions 395
§1.3. Countably Hilbert Spaces and Rigged Hilbert Spaces 398
Appendix II. Background Material on Functional Analysis in Hilbert Spaces 405
§11.1. Operators of Hilbert–Schmidt Type, Nuclear Operators, and Equivalence Operators 405
§11.2. Tensor Products of Hilbert Spaces 415
§11.3. Unitary Representations of Groups 420
Notes and References to the Literature 424
Bibliography 428
Index 432