<STRONG>This book sets out to restructure certain fundamentals in measure and integration theory, and thus to fee the theory from some notorious drawbacks. It centers around the ubiquitous task of producing appropriate contents and measures from more primitive data, in order to extend elementary contents and to represent elementary integrals. This task has not been met with adequate unified means so far. The traditional main tools, the Carathéodory and Daniell-Stone theorems, are too restrictive and had to be supplemented by other ad-hoc procedures. Around 1970 a new approach emerged, based on the notion of regularity, which in traditional measure theory is linked to topology. The present book develops the new approach into a systematic theory. The theory unifies the entire context and is much more powerful than the former means. It has striking implications all over measure theory and beyond. Thus it extends the Riesz representation theorem in terms of Randon measures from locally compact to arbitrary Hausdorff topological spaces. It furthers the methodical unification with non-additive set functions, as shown in natural extensions of the Choquet capacitability theorem. The presentation of this research monograph is self-contained, and starts from the beginning. It is addressed to research workers in mathematical analysis and in applications like mathematical economics, and in particular for university teachers in measure and integration theory. <STRONG>The corrected, second printing includes required corrections and appropriate small alterations of the text and a list of the subsequent articles by the author.
Measure and Integration
Heinz König
Measure and Integration An Advanced Course in Basic Procedures and Applications
Heinz König Universität des Saarlandes Fakultät für Mathematik und Informatik D-66123 Saarbrücken Germany
[email protected]
The cover picture shows a section of the skeleton of a marine diatomee arachnoidiscus. (SEM micrograph reproduced by courtesy of Manfred P. Kage). This is a beautiful example of structure in nature, which reoccurs in man-made buildings and is a model for structure in mathematics. ISBN 978-3-540-61858-4
e-ISBN 978-3-540-89502-2
DOI 10.1007/978-3-540-89502-2 Library of Congress Cataloging-in-Publication Data available. Die Deutsche Bibliothek – CIP-Einheitsaufnahme König, Heinz: Measure and integration: an advanced course in basic procedures and applications/Heinz König. – Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1997 Mathematics Subject Classification (2000): 28-02 Corrected, 2nd printing 2009 c 1997 Springer-Verlag Berlin Heidelberg 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, reproduction on microfilm 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. Violations are liable for prosecution under the German Copyright Law. 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. Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper 987654321 springer.com
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