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The Joys of Haar Measure
Joe Diestel, Kent State University, OH, and Angela Spalsbury, Youngstown State University, OH
 Graduate Studies in Mathematics 2013; 320 pp; hardcover Volume: 150 ISBN-10: 1-4704-0935-6 ISBN-13: 978-1-4704-0935-7 List Price: US$65 Member Price: US$52 Order Code: GSM/150 Not yet published.Expected publication date is April 24, 2014. See also: Introduction to the Mathematics of Finance - R J Williams An Introduction to Measure Theory - Terence Tao The Metric Theory of Tensor Products: Grothendieck's Résumé Revisited - Joe Diestel, Jan H Fourie and Johan Swart 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. Readership Graduate students and research mathematicians interested in measure theory, harmonic analysis, representation theory, and topological groups.