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2013 | Book

Prerequisites Read chapter Read first chapter

Uniform Spaces and Measures

Author: Jan Pachl

Publisher: Springer New York

Book Series : Fields Institute Monographs

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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Until now, results about uniform measures have been scattered through many papers written by a number of authors, some unpublished, written using a variety of definitions and notations.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

Table of Contents

Frontmatter

Uniform Spaces

Frontmatter
Prerequisites
Abstract
The theory developed in this book builds on a number of standard results in topology, functional analysis and measure theory. The prerequisites are summarized in this chapter, along with notation conventions. Definitions of common terms and most proofs are omitted. Detailed explanations and proofs may be found in the reference works cited in each section.
Jan Pachl
Chapter 1. Uniformities and Topologies
Abstract
In this chapter I define uniform structures and uniformly continuous mappings in the language of pseudometrics. I derive their basic properties and their relationship to topologies.
Jan Pachl
Chapter 2. Induced Uniform Structures
Abstract
In this chapter I deal with a frequently used method for constructing uniformities: Given a point-separating set of mappings from a set S to uniform spaces, one uniformity on S is the coarsest for which the mappings are uniformly continuous. This is the induced uniformity on S, also known as the projectively induced or projectively generated uniformity. Special cases of this construction are the uniform subspace, the uniform and semiuniform product and various “weak” uniform structures.
Jan Pachl
Chapter 3. Uniform Structures on Semigroups
Abstract
This chapter begins with basic properties of the right uniformity in semitopological semigroups and in topological groups. Then I abstract the key property of the right uniformity in the definition of a semiuniform semigroup, which will provide a framework for the study of convolution in Chap. 9. I also define ambitable semigroups and prove that many topological groups are ambitable.
Jan Pachl
Chapter 4. Some Notable Classes of Uniform Spaces
Abstract
In this chapter I describe several classes of uniform spaces that will find their use in the study of uniform measures in Parts II and III
Jan Pachl

Uniform Measures

Frontmatter
Chapter 5. Measures on Complete Metric Spaces
Abstract
After several definitions and preliminary results that apply to arbitrary uniform spaces, this chapter deals with tight measures on complete metric spaces.
Jan Pachl
Chapter 6. Uniform Measures
Abstract
When X is a complete metric space, two properties of a functional \(\mathfrak{m}\,\in \,\mathsf{{\mathfrak{M}}_{b}}(X)\) are equivalent, by Lemma 5.2 and Theorem 5.28:
(A)
\(\mathfrak{m}\) is represented by a tight Borel measure on X.
 
(B)
For every Δ ∈ UP(X), the restriction of \(\mathfrak{m}\) to BLip b(Δ) is continuous in the X-pointwise topology.
 
Jan Pachl
Chapter 7. Uniform Measures as Measures
Abstract
In this chapter I discuss the representation of functionals in \(\mathsf{{\mathfrak{M}}_{b}}(X)\) by measures on X and on the uniform compactification \(\widehat{\mathsf{p}}X\).
Jan Pachl
Chapter 8. Instances of Uniform Measures
Abstract
In this chapter I show how to obtain several familiar spaces of measures and measure-like functionals as \(\mathsf{{\mathfrak{M}}_{u}}(X)\) for suitably chosen uniform spaces X, and how to derive properties of spaces of measures from the results in Chap. 6. Although many of the theorems derived here are well known, it is noteworthy that they are all obtained as special cases of general theorems about uniform measures. This brings out parallels between various spaces of measures and indicates to what extent uniform measures are their common generalization.
Jan Pachl
Chapter 9. Direct Product and Convolution
Abstract
As I noted in Sect.6.7, historically one source of the uniform measure concept had been the study of convolution of measures on topological vector spaces and on topological groups. In this chapter I explore the connection between uniform measures and convolution in a fairly general setting that includes convolution on topological groups as a special case.
Jan Pachl

Topics From Farther Afield

Frontmatter
Chapter 10. Free Uniform Measures
Abstract
Uniform measures are functionals on the space of bounded uniformly continuous functions. In this chapter I describe a parallel theory of functionals on the space of all (not necessarily bounded) uniformly continuous functions. The “unbounded version” of \(\mathsf{{\mathfrak{M}}_{u}}(X)\) is the space \(\mathsf{{\mathfrak{M}}_{F}}(X)\) of free uniform measures. The adjective free refers to a universal property of \(\mathsf{{\mathfrak{M}}_{F}}(X)\) that characterizes free functors.
Jan Pachl
Chapter 11. Approximation of Probability Distributions
Abstract
In this chapter, I apply the theory developed in Part II to questions motivated by probability theory. Section 11.1 contains a dual characterization of seminorms ∥ ⋅ ∥ Δ on a large subspace of \(\mathsf{{\mathfrak{M}}_{b}}(X)\). In Sect. 11.2, I discuss properties of asymptotic approximation for nets of probability distributions and apply Corollary P.43 to show that certain notions of approximation are all equivalent on sequences of distributions.
Jan Pachl
Chapter 12. Measurable Functionals
Abstract
In this chapter, I describe an approach to automatic continuity of functionals on U b(X): In order to prove that a linear functional \(\mathfrak{m}\) on U b(X) is a uniform measure, it is sometimes enough to prove that \(\mathfrak{m}\) is measurable with respect to a suitable σ-algebra on U b(X). Measurability is often easier to establish than continuity; this is illustrated by the results in Sect. 12.3.
Jan Pachl
Backmatter
Metadata
Title
Uniform Spaces and Measures
Author
Jan Pachl
Copyright Year
2013
Publisher
Springer New York
Electronic ISBN
978-1-4614-5058-0
Print ISBN
978-1-4614-5057-3
DOI
https://doi.org/10.1007/978-1-4614-5058-0

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