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Integration – a Functional Approach

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Integration – a Functional Approach Klaus Bichteler Integration – A Functional Approach Reprint of the 1998 Edition Klaus Bichteler Department of Mathematics The University of Texas Austin, TX 78712 USA [email protected] 1991 Mathematics Subject Classification 28-01, 28C05 ISBN 978-3-0348-0054-9 e-ISBN 978-3-0348-0055-6 DOI 10.1007/978-3-0348-0055-6 c 1998 Birkhauser¨ Verlag Originally published under the same title in the Birkhauser¨ Advanced Texts – Basler Lehrbucher¨ series by Birkhauser¨ Verlag, Switzerland, ISBN 978-3-7643-5936-2 Reprinted 2010 by Springer Basel AG This work is subject to copyright. All rights are reserved, whether the whole or part of the material is con- cerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com For Ursula Contents Preface ............................................................. ix Chapter I Review ...................................................... 1 I.1 Introduction ..................................................... 1 I.2 Notation ........................................................ 3 I.3 The Theorem of Stone–Weierstraß ............................... 8 I.4 The Riemann Integral .......................................... 17 I.5 An Integrability Criterion ...................................... 21 I.6 The Permanence Properties .................................... 23 I.7 Seminorms ..................................................... 28 Chapter II Extension of the Integral .................................. 31 II.1 Σ–additivity ................................................... 33 II.2 Elementary Integrals ........................................... 35 II.3 The Daniell Mean .............................................. 42 II.4 Negligible Functions and Sets ................................... 48 II.5 Integrable Functions ............................................ 52 II.6 Extending the Integral ......................................... 58 II.7 Integrable Sets ................................................. 61 II.8 Example of a Non–Integrable Function .......................... 66 Chapter III Measurability ............................................. 69 III.1 Littlewood’s Principles ......................................... 70 III.2 The Permanence Properties .................................... 73 III.3 The Integrability Criterion ..................................... 75 III.4 Measurable Sets ................................................ 81 III.5 Baire and Borel Functions ...................................... 85 III.6 Further Properties of Daniell’s Mean ............................ 92 III.7 The Procedures of Lebesgue and Carath´edory ................... 95 viii Contents Chapter IV The Classical Banach Spaces ............................. 101 IV.1 The p–norms .................................................. 102 IV.2 The Lp –spaces ................................................ 105 IV.3 The Lp–spaces ................................................ 108 IV.4 Linear Functionals ............................................ 112 IV.5 The Dual of Lp ............................................... 118 IV.6 The Hilbert space L2 .......................................... 122 Chapter V Operations on Measures .................................. 125 V.1 Products of Elementary Integrals .............................. 125 V.2 The Theorems of Fubini and Tonelli ........................... 130 V.3 An Application: Convolution .................................. 133 V.4 An Application: Marcinkiewicz Interpolation ................... 134 V.5 Signed Measures .............................................. 137 V.6 The Space of Measures ........................................ 144 V.7 Measures with Densitites ...................................... 151 V.8 The Radon–Nikodym Theorem ................................ 154 V.9 An Application: Conditional Expectation ...................... 155 V.10 Differentiation ................................................ 157 Appendix A Answers to Selected Problems ........................... 165 References ......................................................... 180 Index of Notations ................................................. 181 Index .............................................................. 184 Preface This text originated as notes for the introductory graduate Real Analysis course at the University of Texas at Austin. They were intended for students who have had a first course in Real Analysis (−δ–proofs) and know the topology of the real line, in particular the notion of compactness, and the Riemann integral. The main topic in such a course traditionally is integration theory, and so it is here. The approach taken is different from the usual one, though. Usually the development follows history. First the content is extended to a measure on a σ–algebra, — which is identified through Carath´eodory’s cut condition — then to the integrable functions through another set of limit operations. Daniell observed that the extensions have much in common and can be done in one effort, saving half the labor. We follow in his footsteps. The emphasis is not, however, on saving time so much as it is on functionality: we ask what the purpose and expected properties of the prospective integral are and then fashion definitions and procedures accordingly—history and the usual treatment are reviewed, though, in Section III.7, so as to give the reader the connection of the historical development with the present one. Functionality also guides the treatment of measurability. The text does not set down the notion of a σ–algebra as a gift from our betters, followed by a (mysterious) definition of measurability on them; this would pressure the gentle reader into accepting authoritarian mathematics or leave the rebellious one with many questions: where do these definitions come from, where do they lead, and how did anyone ever dream them up? Rather, we ask what the obstruction to the integrability of a function might be besides being too big, in other words, what the local structure of the integrable functions is. Two simple observations in answer to this lead to Littlewood’s Principles, which are used to define measurability. The permanence properties of measurability are rather easy to establish this way, and once they are available so is the fact that the definition agrees with the traditional one. A third uncommon aspect of the development is the use of seminorms. Instead of arguing with the upper and lower integrals of Daniell, analogs of Lebesgue’s outer and inner measure, we observe that already in the case of the Riemann integral a simple absolute–value sign under the upper integral simplifies matters considerably, by rendering superfluous the study of the lower integral and pro- viding a seminorm, the Daniell mean. The analyst will often attack a problem by defining a suitable seminorm with which to identify a function space in which a solution reasonably can be sought. It seems to me that it is not too early to introduce this tool in the context of integration: the problem is to extend the ele- mentary integral from step functions to larger classes, and the Jordan and Daniell means are perfectly suitable seminorms towards that goal. x Preface The exposition starts with a review of Weierstraß’ theorem and the Riemann integral. The latter review is designed to show that the integral of Lebesgue– Carath´eodory emerges from rather a minute alteration to Riemann’s; it is not the mystery but the plethora of wonderful properties of the new integral that ask for another 120 pages just to write them down and prove them. The exercises are an integral part of the material. Those marked by an asterisk ∗ are used later on and thus must be done. For the majority of them there are answers or at least hints in the appendix. The first version of these notes was replete with mistakes, and I thank my students for pointing them out. I hope the remaining mistakes are small in number; they are all mine. Klaus Bichteler The University of Texas at Austin //www.ma.utexas.edu/users/kbi [email protected] May 1998 Chapter I Review I.1 Introduction Riemann’s integral is a major achievement of civilization. It solves in a rigourous way the ancient problems of squaring the disk and straightening the circle. It provides a solid foundation for Newtonian Physics, on which modern theories of physics and engineering can grow. Nevertheless, it has shortcomings: its limit theorems are not good enough, and not sufficiently many functions are Riemann– integrable. Consequently it is an insufficient tool for quantum physics and the associated partial differential equations. These shortcomings can be overcome, though, by a careful analysis of its concepts and a surprisingly slight improvement on them. The integral of a step function whose steps are intervals is the sum of the products step–size×height–of–step. Riemann’s well–known procedure to integrate a function f that is not a step function is to squeeze it with step–functions from above and below: One defines the upper Riemann integral of f to be the infimum of the integrals of step–functions above f and its lower Riemann integral to be the supremum of the integrals of step–functions below. Then one says that f is Riemann integrable if these two numbers
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