Table Of Content

Principles of Analysis Principles of Analysis Measure, Integration, Functional Analysis, and Applications Hugo D. Junghenn CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20180324 International Standard Book Number-13: 978-1-4987-7328-7 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Junghenn, Hugo D. (Hugo Dietrich), 1939- author. Title: Principles of real analysis : measure, integration, functional analysis, and applications / Hugo D. Junghenn. Description: Boca Raton : CRC Press, Taylor & Francis Group, 2018. | Includes bibliographical references and index. Identifiers: LCCN 2017061660 | ISBN 9781498773287 Subjects: LCSH: Functions of real variables--Textbooks. | Mathematical analysis--Textbooks. Classification: LCC QA331.5 .J86 2018 | DDC 515/.8--dc23 LC record available at https://lccn.loc.gov/2017061660 Visit the e-resources at: https://www.crcpress.com/9781498773287 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com TO MY CHILDREN Katie and Patrick AND TO MY WIFE Mary AS ALWAYS Contents Preface xix 0 Preliminaries 1 0.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.2 Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Semigroups and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Quotient Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 0.3 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Interior, Closure, and Boundary . . . . . . . . . . . . . . . . . . . . . . . 12 Sequential Convergence. Completeness . . . . . . . . . . . . . . . . . . . 12 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 0.4 Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Norms and Seminorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Completion of a Normed Space . . . . . . . . . . . . . . . . . . . . . . . 16 Infinite Series in Normed Spaces . . . . . . . . . . . . . . . . . . . . . . 16 Unordered Sums in Normed Spaces . . . . . . . . . . . . . . . . . . . . . 17 Bounded Linear Transformations . . . . . . . . . . . . . . . . . . . . . . 18 Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 0.5 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Neighborhood Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Neighborhood Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Relative Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 0.6 Continuity in Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . 23 Definition and General Properties . . . . . . . . . . . . . . . . . . . . . . 23 Initial Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Product Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Final Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Quotient Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 vii viii Contents The Space of Continuous Functions . . . . . . . . . . . . . . . . . . . . . 25 F-sigma and G-delta Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 25 0.7 Normal Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Urysohn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Tietze Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0.8 Compact Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 27 Convergence in Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . 28 Compactness of Cartesian Products . . . . . . . . . . . . . . . . . . . . . 29 Continuity and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 29 0.9 Totally Bounded Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . 30 0.10 Equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0.11 The Stone-Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . . . . 32 0.12 Locally Compact Topological Spaces . . . . . . . . . . . . . . . . . . . . . 33 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Functions with Compact Support . . . . . . . . . . . . . . . . . . . . . . 34 Functions That Vanish at Infinity . . . . . . . . . . . . . . . . . . . . . . 35 The One-Point Compactification . . . . . . . . . . . . . . . . . . . . . . . 35 0.13 Spaces of Differentiable Functions . . . . . . . . . . . . . . . . . . . . . . 36 0.14 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 0.15 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 I Measure and Integration 41 1 Measurable Sets 43 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.2 Measurable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Fields and Sigma Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Generated Sigma Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Extended Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Product Sigma Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Pi-Systems and Lambda-Systems . . . . . . . . . . . . . . . . . . . . . . 47 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.3 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Set Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Properties and Examples of Measures . . . . . . . . . . . . . . . . . . . . 51 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.4 Complete Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Completion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Null Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.5 Outer Measure and Measurability . . . . . . . . . . . . . . . . . . . . . . 55 Construction of an Outer Measure . . . . . . . . . . . . . . . . . . . . . 56 Carath´eodory’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.6 Extension of a Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 The Measure Extension Theorem . . . . . . . . . . . . . . . . . . . . . . 59 Approximation Property of the Extension . . . . . . . . . . . . . . . . . . 61 Completeness of the Extension . . . . . . . . . . . . . . . . . . . . . . . . 61 Uniqueness of the Extension . . . . . . . . . . . . . . . . . . . . . . . . . 62 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Contents ix 1.7 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 The Volume Set Function . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Construction of the Measure . . . . . . . . . . . . . . . . . . . . . . . . . 65 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 1.8 Lebesgue-Stieltjes Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 One-Dimensional Distribution Functions . . . . . . . . . . . . . . . . . . 67 ∗Higher Dimensional Distribution Functions . . . . . . . . . . . . . . . . . 69 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 *1.9 Some Special Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 An Uncountable Set with Lebesgue Measure Zero . . . . . . . . . . . . . . 71 Non-Lebesgue-Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . . 71 A Lebesgue Measurable, Non-Borel Set . . . . . . . . . . . . . . . . . . . 72 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2 Measurable Functions 75 2.1 Measurable Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 75 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.2 Measurable Numerical Functions . . . . . . . . . . . . . . . . . . . . . . . 78 Criteria for Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Almost Everywhere Properties . . . . . . . . . . . . . . . . . . . . . . . . 79 Combinatorial and Limit Properties of Measurable Functions . . . . . . . 79 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.3 Simple Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A Fundamental Convergence Theorem . . . . . . . . . . . . . . . . . . . . 82 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.4 Convergence of Measurable Functions . . . . . . . . . . . . . . . . . . . . 85 Modes of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Relationships Among the Modes of Convergence . . . . . . . . . . . . . . 86 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3 Integration 89 3.1 Construction of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Integral of a Nonnegative Simple Function . . . . . . . . . . . . . . . . . 89 Integral of a Real-Valued Function . . . . . . . . . . . . . . . . . . . . . . 90 Integral of a Complex-Valued Function . . . . . . . . . . . . . . . . . . . 91 Integral over a Measurable Set . . . . . . . . . . . . . . . . . . . . . . . . 91 3.2 Basic Properties of the Integral . . . . . . . . . . . . . . . . . . . . . . . . 92 Almost Everywhere Properties . . . . . . . . . . . . . . . . . . . . . . . . 92 Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . 93 Linearity of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Integration Against an Image Measure . . . . . . . . . . . . . . . . . . . 96 Integration Against a Measure with Density . . . . . . . . . . . . . . . . . 96 Change of Variables Theorem . . . . . . . . . . . . . . . . . . . . . . . . 97 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.3 Connections with the Riemann Integral on Rd . . . . . . . . . . . . . . . 100 The Darboux Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Measure Zero Criterion for Riemann Integrability . . . . . . . . . . . . . 104

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Principles of Analysis: Measure, Integration, Functional Analysis, and Applications prepares readers taking advanced courses in analysis, probability, harmonic analysis, and applied mathematics at the doctoral level. It is also designed so that the reader or instructor may select topics suitable to
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