本书旨在阐述拓扑群上的不变测度，从特殊情形逐步过渡到更一般的情况。本书适合已修读实变函数基础课程的研究生和高年级本科生阅读。 From the earliest days of measure theory, invariant measures have held the interests of geometers and analysts alike, with the Haar measure playing an especially delightful role. The aim of this book is to present invariant measures on topological groups, progressing from special cases to the more general. Presenting existence proofs inspecial cases, such as compact metrizable groups, highlights how the added assumptions give insight into just what the Haar measure is like; tools from different aspects of analysis and/or combinatorics demonstrate the diverse views afforded the subject. After presenting the compact case, applications indicate how these tools can find use.

前辅文
Preface
Chapter 1. Lebesgue Measure in Euclidean Space
  §1. An Introduction to Lebesgue Measure
  §2. The Brunn–Minkowski Theorem
  §3. Covering Theorem of Vitali
  §4. Notes and Remarks
Chapter 2. Measures on Metric Spaces
  §1. Generalities on Outer Measures
  §2. Regularity
  §3. Invariant Measures on Rn
  §4. Notes and Remarks
Chapter 3. Introduction to Topological Groups
  §1. Introduction
  §2. The Classical (Locally Compact) Groups
  §3. The Birkhoff–Kakutani Theorem
  §4. Products of Topological Spaces
  §5. Notes and Remarks
Chapter 4. Banach and Measure
  §1. Banach Limits
  §2. Banach and Haar Measure
  §3. Saks’ Proof of C(Q)*, Q a Compact Metric Space
  §4. The Lebesgue Integral on Abstract Spaces
  §5. Notes and Remarks
Chapter 5. Compact Groups Have a Haar Measure
  §1. The Arzelá–Ascoli Theorem
  §2. Existence and Uniqueness of an Invariant Mean
  §3. The Dual of C(K)
  §4. Notes and Remarks
Chapter 6. Applications
  §1. Homogeneous Spaces
  §2. Unitary Representations: The Peter–Weyl Theorem
  §3. Pietsch Measures
  §4. Notes and Remarks
Chapter 7. Haar Measure on Locally Compact Groups
  §1. Positive Linear Functionals
  §2. Weil’s Proof of Existence
  §3. A Remarkable Approximation Theorem of Henri Cartan
  §4. Cartan’s Proof of Existence of a Left Haar Integral
  §5. Cartan’s Proof of Uniqueness
  §6. Notes and Remarks
Chapter 8. Metric Invariance and Haar Measure
  §1. Notes and Remarks
Chapter 9. Steinlage on Haar Measure
  §1. Uniform Spaces: The Basics
  §2. Some Miscellaneous Facts and Features about Uniform Spaces
  §3. Compactness in Uniform Spaces
  §4. From Contents to Outer Measures
  §5. Existence of G-invariant Contents
  §6. Steinlage: Uniqueness and Weak Transitivity
  §7. Notes and Remarks
Chapter 10. Oxtoby’s View of Haar Measure
  §1. Invariant Measures on Polish Groups
  §2. Notes and Remarks
Appendix A
Appendix B
Bibliography
Author Index
Subject Index