Uniform Spaces and Measures

1. Front Matter

   Pages i-ix

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2. Uniform Spaces

   1. Front Matter

      Pages 17-17

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   2. Prerequisites

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      Pages 1-16

   3. Uniformities and Topologies

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      Pages 19-28

   4. Induced Uniform Structures

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      Pages 29-39

   5. Uniform Structures on Semigroups

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      Pages 41-51

   6. Some Notable Classes of Uniform Spaces

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      Pages 53-60

3. Uniform Measures

   1. Front Matter

      Pages 61-61

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   2. Measures on Complete Metric Spaces

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      Pages 63-79

   3. Uniform Measures

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      Pages 81-98

   4. Uniform Measures as Measures

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      Pages 99-116

   5. Instances of Uniform Measures

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      Pages 117-127

   6. Direct Product and Convolution

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      Pages 129-147

4. Topics From Farther Afield

   1. Front Matter

      Pages 149-149

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   2. Free Uniform Measures

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      Pages 151-169

   3. Approximation of Probability Distributions

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      Pages 171-177

   4. Measurable Functionals

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      Pages 179-185

5. Back Matter

   Pages 187-209

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​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Until now, results about uniform measures have been scattered through many papers written by a number of authors, some unpublished, written using a variety of definitions and notations.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.​

• abstract harmonic analysis
• semitopological semigroups
• topologies on spaces of measures
• uniform measures

From the reviews:

“The material of this book is arranged into one preliminary section for auxiliary prerequisites plus three parts. … Each of these parts is divided into chapters, every one of which ends with several complementary and historical notes that clearly help one to gain a better understanding of the material being covered. … this is a book which every researcher interested in measure theory and its applications must consider for library acquisition and personal use.” (Jesús Ferrer, Mathematical Reviews, August, 2013)

• The Fields Institute for Research in Mat, Toronto, Canada

  Jan Pachl

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