This book addresses the need for an accessible comprehensive exposition of the theory of uniform measures; the need that became more critical when recently uniform measures reemerged in new results in abstract harmonic analysis. Until now, results about uniform measures have been scattered through many papers written by a number of authors, some unpublished, written using a variety of definitions and notations. Uniform measures are certain functionals on the space of bounded uniformly continuous functions on a uniform space. They are a common generalization of several classes of measures and measure-like functionals studied in abstract and topological measure theory, probability theory, and abstract harmonic analysis. They offer a natural framework for results about topologies on spaces of measures and about the continuity of convolution of measures on topological groups and semitopological semigroups. The book is a reference for the theory of uniform measures. It includes a self-contained development of the theory with complete proofs, starting with the necessary parts of the theory of uniform spaces. It presents diverse results from many sources organized in a logical whole, and includes several new results. The book is also suitable for graduate or advanced undergraduate courses on selected topics in topology and functional analysis. The text contains a number of exercises with solution hints, and four problems with suggestions for further research.
Author(s): Jan Pachl
Series: Fields Institute Monographs, Vol. 30
Publisher: Springer
Year: 2013
Language: English
Pages: 221
Cover......Page 1
Title Page......Page 5
Preface......Page 7
Contents......Page 9
P.1 Sets and Mappings......Page 13
P.2 Topological Spaces and Groups......Page 14
P.3 Topological Vector Spaces......Page 17
P.4 Riesz Spaces......Page 20
P.5 Measures......Page 22
P.6 Prerequisites for Part III......Page 27
Part I: Uniform Spaces......Page 29
1.1 Uniform Structures and Mappings......Page 31
1.2 Cardinal Reflections, Compactness and Completeness......Page 34
1.3 Metric Spaces and Real Functions......Page 36
1.4 Uniformizable Topological Spaces......Page 39
1.5 Notes for Chap. 1......Page 40
2.1 General Properties......Page 41
2.2 Uniform Subspaces......Page 45
2.3 Uniform Structures on Products......Page 48
2.4 Notes for Chap. 2......Page 51
3.1 Semitopological Semigroups......Page 53
3.2 Topological Groups......Page 55
3.3 Semiuniform Semigroups......Page 56
3.4 Ambitable Groups......Page 58
3.5 Notes for Chap. 3......Page 62
4.1 Inversion-Closed and Alexandroff Spaces......Page 65
4.2 Supercomplete Spaces......Page 69
4.3 Partitions of Unity......Page 71
4.4 Notes for Chap. 4......Page 72
Part II Uniform Measures......Page 73
5.1 Tight Measures on Uniform Spaces......Page 75
5.2 Point Masses and Molecular Measures......Page 77
5.3 Two Weak Topologies......Page 78
5.4 Tight Measures on Complete Metric Spaces......Page 80
5.5 Topologies on the Positive Cone and on Spheres......Page 85
5.6 Compact Sets and Sequences of Measures......Page 87
5.7 Notes for Chap. 5......Page 90
6 Uniform Measures......Page 93
6.1 Space of Uniform Measures......Page 94
6.2 Image Under a Uniformly Continuous Mapping......Page 96
6.3 Topologies on the Space of Uniform Measures......Page 98
6.4 Uniform Measures on Induced Spaces......Page 100
6.5 Completion and Compactification......Page 105
6.6 Vector-Valued Integrals......Page 107
6.7 Vector-Valued Uniform Measures......Page 108
6.8 Notes for Chap. 6......Page 109
7.1 Functionals Represented by Measures......Page 111
7.2 Measures on the Compactification......Page 114
7.3 Conditions for Uniform Measures to be Measures......Page 117
7.4 Condition for Measures to be Uniform Measures......Page 120
7.5 Measure-Fine Uniform Spaces......Page 121
7.6 Notes for Chap. 7......Page 126
8.1 Sequentially Uniform Measures......Page 129
8.2 Measures on Abstract σ-Algebras......Page 131
8.3 Baire Measures on Completely Regular Spaces......Page 133
8.4 Separable Measures......Page 135
8.5 Tight Measures on Locally Compact Groups......Page 137
8.7 Notes for Chap. 8......Page 138
9.1 Direct Product......Page 141
9.2 Convolution for Semiuniform Semigroups......Page 147
9.3 Topological Centres in Convolution Semigroups......Page 150
9.4 Case of Topological Groups......Page 153
9.5 Notes for Chap. 9......Page 157
Part III Topics from Farther Afield......Page 161
10.1 Basic Properties......Page 163
10.2 Universal Property......Page 171
10.3 Measures with Compact Support......Page 173
10.5 Measures on Abstract σ-Algebras......Page 176
10.6 Riesz Measures on Completely Regular Spaces......Page 177
10.7 Cylindrical Measures of Type 1......Page 179
10.8 Notes for Chap. 10......Page 180
11.1 The Kantorovich–Rubinshteĭn Theorem......Page 183
11.2 Asymptotic Approximation of Probability Distributions......Page 186
11.3 Notes for Chap. 11......Page 189
12.1 Saturated Spaces......Page 191
12.2 Functionals of Baire Class 1......Page 193
12.3 Generalized Centres in Convolution Semigroups......Page 195
12.4 Notes for Chap. 12......Page 196
Hints to Exercises......Page 199
References......Page 205
Notation Index......Page 213
Author Index......Page 215
Subject Index......Page 219
