Radon Integrals An abstract approach to integration and Riesz repres
In topological measure theory, Radon measures are the most important objects. In the context of locally compact spaces, there are two equivalent canonical definitions. As a set function, a Radon measure is an inner compact regular Borel measure, finite on
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Series Editors J. Oesterle A. Weinstein
BemdAnger Claude Portenier
Radon Integrals An abstract approach to integration and Riesz representation throughfunction cones
Springer Science+Business Media, LLC
BemdAnger Mathematisches Institut Universitllt Erlangen-Niimberg 8520 Erlangen Germany
Claude Portenier Fachbereich Mathematik Universitllt Marburg 3550 Marburg Germany
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© Springer Science+Business Media New York 1992. Originally published by Birkhiiuser Boston in 1992 Softcover reprint ofthe hardcover 1st edition 1992 AlI rights reserved. No part of this publication may be reproduced, s10red in a retrieval system,or transmitted,in any formorby any means,electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission 10 pho1ocopy for internal or personal use of specific clients is granted by Springer Science+Business Media, LLC, for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per page is paid directly 10 CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Springer Science+Business Media, LLC.
ISBN 978-1-4612-6733-1 ISBN 978-1-4612-0377-3 (eBook) DOI 10.1007/978-1-4612-0377-3 Camera-ready text prepared by the Authors.
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PREFACE
In topological measure theory, Radon measures are the most important objects. In the context of locally compact spaces, there are two equivalent canonical definitions. As a set function, a Radon measure is an inner compact regular Borel measure, finite on compact sets. As a functional, it is simply a positive linear form, defined on the vector lattice of continuous real-valued functions with compact support. During the last few decades, in particular because of the developments of modem probability theory and mathematical physics, attention has been focussed on measures on general topological spaces which are no longer locally compact, e.g. spaces of continuous functions or Schwartz distributions. For a Radon measure on an arbitrary Hausdorff space, essentially three equivalent definitions have been proposed: As a set function, it was defined by L. Schwartz as an inner compact regular Borel measure which is locally bounded. G. Choquet considered it as a strongly additive right continuous content on the lattice of compact subsets. Following P.A. Meyer, N. Bourbaki defined a Radon measure as a locally uniformly bounded family of compatible positive linear forms, each defined on the vector lattice of continuous functions on some compact subset. Compared with the simplicity of the functional analytic description in the locally compact case, it seems that the "linear functional aspect" of Radon measures has been lost in the general situation. It is our aim to show how to define "Radon integrals" as certain linear functionals, and then how to develop a theory of integration in a functional analytic spirit. Obviously, the vector lattice of continuous functio
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