Lecture Notes in Mathematics

Transcription

1 Lecture Notes in Mathematics Edited by J.-M. Morel, F. Takens and B. Teissier Editorial Policy for the publication of monographs l. Lecture Notes aim to report new developments in all areas of mathematics - quickly, informally and at a high level. Monograph manuscripts should be reasonably self-contained and rounded off. Thus they may, and often will, present not only results of the author but also related work by other people. They may be based on specialized lecture courses. Furthermore, the manuscripts should provide sufficient motivation, examples and applications. This clearly distinguishes Lecture Notes from journal articles or technical reports which normally are very concise. Articles intended for a journal but too long to be accepted by most journals, usually do not have this "lecture notes" character. For similar reasons it is unusual for doctoral theses to be accepted for the Lecture Notes series. 2. Manuscripts should be submitted (preferably in duplicate) either to one of the series editors or to Springer-Verlag, Heidelberg. In general, manuscripts will be sent out to 2 external referees for evaluation. If a decision cannot yet be reached on the basis of the first 2 reports, further referees may be contacted: the author will be informed of this. A final decision. to publish can be made only on the basis of the complete manuscript, however a refereeing process leading to a preliminary decision can be based on a pre-final or incomplete manuscript. The strict minimum amount of material that will be considered should include a detailed outline describing the planned contents of each chapter, a bibliography and several sample chapters. Authors should be aware that incomplete or insufficiently close to final manuscripts almost always result in longer refereeing times and nevertheless unclear referees' recommendations, making further refereeing of a final draft necessary. Authors should also be aware that parallel submission of their manuscript to another publisher while under consideration for LNM will in general lead to immediate rejection. 3. Manuscripts should in general be submitted in English. Final manuscripts should contain at least 100 pages of mathematical text and should include - a table of contents; - an informative introduction, with adequate motivation and perhaps some historical remarks: it should be accessible to a reader not intimately familiar with the topic treated; - a subject index: as a rule this is genuinely helpful for the reader. Continued on inside back-cover

2 Lecture Notes in Mathematics 1808 Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

3 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

4 Werner Schindler Measures with Symmetry Properties 13

5 Author Werner Schindler Bundesamt für Sicherheit in der Informationstechnik (BSI) Godesberger Allee Bonn, Germany 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 Mathematics Subject Classification (2000): 28C10, 65C10, 62B05, 22E99 ISSN ISBN Springer-VerlagBerlinHeidelbergNewYork This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsofthegermancopyrightlawofseptember9, 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 c Springer-Verlag Berlin Heidelberg 2003 PrintedinGermany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX outputbytheauthor SPIN: /3142/du Printed on acid-free paper

6 In memory of my father

7 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 IR n 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 mathematics with which we have to cope to be deep and hard. For instance, certain techniques which have been very efficient in differential geometry are not available to us. On the other hand, the generality of our approach allows us to apply it to an amazingly wide range of problems. This new scope of applications is the positive side of the generality of our attack. The main results of this book belong to measure and integration theory and thus to pure mathematics. However, in the second part of the book many concrete applications are discussed in great detail. We consider integration problems, stochastic simulations and statistics. The examples span from computational geometry, applied mathematics, computer aided graphical design to coding theory. This book therefore does not only address pure and applied mathematicians, but also computer scientists and engineers. As a consequence, I have attempted to make the discourse accessible to readers from a great variety of different backgrounds and to keep the prerequisites to a minimum. Only in Chapter 2 where the core theorems are derived did it appear impossible to adhere strictly to this goal. However, I have arranged the presentation in such a fashion that a reader who wishes to skip this chapter completely, can read the remaining parts of the book without loss of under-

8 VIII Preface standing. As far as they are relevant for the applications, the main results of Chapter 2 are summarized at the beginning of Chapter 4. This book arose from my postdoctoral thesis (Habilitationsschrift, [71]). I would like to thank Karl Heinrich Hofmann, Jürgen Lehn and Gunter Ritter again for acting as referees and for their valuable advice, Siegfried Graf for providing me with an interesting reference, and Hans Tzschach for the encouragement he has given me to even consider the writing of this thesis. Relative to the postdoctoral thesis, in the present book I have added some new results and a number of illustrative examples. Of course there are editorial adjustments, and the text has been made more self-contained ; this should enhance its readability without forcing the reader to consult too many references. I would like to thank Karl Heinrich Hofmann for his advice and support. Finally, I want to thank my wife Susanne for her patience with me when I was writing my postdoctoral thesis and this book. Sinzig, October 2002 Werner Schindler

9 Table of Contents 1 Introduction Main Theorems Definitions and Preparatory Lemmata Definition of Property ( ) and Its Implications (Main Results) Supplementary Expositions and an Alternate Existence Proof 43 3 Significance, Applicability and Advantages Applications Central Definitions, Theorems and Facts Equidistribution on the Grassmannian Manifold and Chirotopes Conjugation-invariant Probability Measures on Compact Connected Lie Groups Conjugation-invariant Probability Measures on SO(n) Conjugation-invariant Probability Measures on SO(3) The Theorem of Iwasawa and Invariant Measures on Lie Groups QR-Decomposition on GL(n) Polar Decomposition on GL(n) O(n)-invariant Borel measures on Pos(n) Biinvariant Borel Measures on GL(n) Symmetries on Finite Spaces References Glossary Index