The Joys of Haar Measure (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 150)

"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 in special 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. The generalization to locally compact groups is then presented and applied to show relations between metric and measure theoretic invariance. Steinlage's approach to the general problem of homogeneous action in the locally compact setting shows how Banach's approach and that of Cartan and Weil can be unified with good effect. Finally, the situation of a nonlocally compact Polish group is discussed briefly with the surprisingly unsettling consequences indicated. The book is accessible to graduate and advanced undergraduate students who have been exposed to a basic course in real variables, although the authors do review the development of the Lebesgue measure. It will be a stimulating reference for students and professors who use the Haar measure in their studies and research."--Contratapa. 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 In Special 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. The Generalisation To Locally Compact Groups Is Then Presented And Applied To Show Relations Between Metric And Measure Theoretic Invariance. Steinlage's Approach To The General Problem Of Homogeneous Action In The Locally Compact Setting Shows How Banach's Approach And That Of Cartan And Weil Can Be Unified With Good Effect. Finally, The Situation Of A Nonlocally Compact Polish Group Is Discussed Briefly With The Surprisingly Unsettling Consequences Indicated. The Book Is Accessible To Graduate And Advanced Undergraduate Students Who Have Been Exposed To A Basic Course In Real Variables, Although The Authors Do Review The Development Of The Lebesgue Measure. It Will Be A Stimulating Reference For Students And Professors Who Use The Haar Measure In Their Studies And Research. --provided By Publisher. Lebesgue Measure In Euclidean Space -- Measures On Metric Spaces -- Introduction To Topological Groups -- Banach And Measure -- Compact Groups Have A Haar Measure -- Applications -- Haar Measure On Locally Compact Groups -- Metric Invariance And Haar Measure -- Steinlage On Haar Measure -- Oxtoby's View Of Haar Measure. Joe Diestel, Angela Spalsbury. Includes Bibliographical References And Index. Preface -- 1. Lebesgue Measure in Euclidean Space -- 2. Measures on Metric Spaces -- 3. Introduction to Topological Groups -- 4. Banach and Measure -- 5. Compact Groups Have a Haar Measure -- 6. Applications -- 7. Haar Measure on Locally Compact Groups -- 8. Metric Invariance and Haar Measure -- 9. Steinlage on Haar Measure -- 10. Oxtoby's View of Haar Measure.