Measures with Symmetry Properties
Symmetries and invariance principles play an important role in various branches of mathematics. This book deals with measures having weak symmetry properties. Even mild conditions ensure that all invariant Borel measures on a second countable locally comp
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Werner Schindler
Measures with Symmetry Properties
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Author Werner Schindler Bundesamt f¨ur Sicherheit in der Informationstechnik (BSI) Godesberger Allee 185-189 53175 Bonn, Germany E-mail: [email protected]
Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): 28C10, 65C10, 62B05, 22E99 ISSN 0075-8434 ISBN 3-540-00235-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10825787
41/3142/du-543210 - Printed on acid-free paper
In memory of my father
Preface
Symmetry has played an important role in art and architecture since early mankind. Symmetries occur in botany, biology, chemistry and physics. It is therefore not surprising that in various branches of mathematics symmetries and invariance principles have emerged and were put to good use as research tools. A prominent example is measure and integration theory where a rich theory has been developed for measures with remarkable symmetry properties such as Haar measures on locally compact groups, invariant measures on homogeneous spaces, or particular measures on IRn with special symmetries. In connection with invariant statistics research has focussed on specific situations in which measures that do not have ‘full’ symmetry occur. This book also deals with measures having symmetry properties. However, our assumptions on the symmetry properties and on the general framework in which we are working are rather mild. This has two somewhat contradictory consequences. On the one hand, the weakness of the hypotheses we impose and the ensuing generality cause the mathematic
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