ebook img

Elementary introduction to the Lebesgue integral PDF

197 Pages·2018·0.92 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Elementary introduction to the Lebesgue integral

Elementary Introduction to the Lebesgue Integral Textbooks in Mathematics Series editors: Al Boggess and Ken Rosen A TOUR THROUGH GRAPH THEROY Karin R. Saoub TRANSITION TO ANALYSIS WITH PROOF Steven Krantz ESSENTIALS OF MATHEMATICAL THINKING Steven G. Krantz ELEMENTARY DIFFERENTIAL EQUATIONS Kenneth Kuttler A CONCRETE INTRODUCTION TO REAL ANALYSIS, SECOND EDITION Robert Carlson MATHEMATICAL MODELING FOR BUSINESS ANALYTICS William Fox ELEMENTARY LINEAR ALGEBRA James R. Kirkwood APPLIED FUNCTIONAL ANALYSIS, THIRD EDITION J. Tinsley Oden AN INTRODUCTION TO NUMBER THEORY WITH CRYPTOGRAPHY, SECOND EDITION James Kraft MATHEMATICAL MODELING: BRANCHING BEYOND CALCULUS Crista Arangala ELEMENTARY DIFFERENTIAL EQUATIONS, SECOND EDITION Charles Roberts ELEMENTARY INTRODUCTION TO THE LEBESGUE INTEGRAL Steven G. Krantz Elementary Introduction to the Lebesgue Integral Steven G. Krantz 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: 20180212 International Standard Book Number-13: 978-1-138-48276-0 (Paperback) 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: Krantz, Steven G. (Steven George), 1951- author. Title: An introduction to the Lebesgue integral / Steven G. Krantz. Description: Boca Raton : CRC Press, Taylor & Francis Group, 2018. | Includes bibliographical references and index. Identifiers: LCCN 2017061661 | ISBN 9781138482760 Subjects: LCSH: Lebesgue integral--Textbooks. | Integrals, Generalized--Textbooks. | Measure theory--Textbooks. Classification: LCC QA312 .K7325 2018 | DDC 515/.43--dc23 LC record available at https://lccn.loc.gov/2017061661 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To the memory of Phil Curtis, a great mentor and friend. Contents Preface xi 1 Introductory Thoughts 1 1.1 Review of the Riemann Integral . . . . . . . . . . . . . . . . 1 1.2 The Idea of the Lebesgue Integral . . . . . . . . . . . . . . 4 1.3 Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . . 8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 The Purpose of Measures 19 2.1 What Is a Measure? . . . . . . . . . . . . . . . . . . . . . . 19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 The Lebesgue Integral 25 3.1 The Integration Theory of Lebesgue . . . . . . . . . . . . . 25 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Integrable Functions 35 4.1 Functions with Finite Integral . . . . . . . . . . . . . . . . . 35 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5 The Lebesgue Spaces 43 5.1 Definition of the Spaces . . . . . . . . . . . . . . . . . . . . 43 5.2 The Case p= . . . . . . . . . . . . . . . . . . . . . . . . 49 ∞ Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6 The Concept of Outer Measure 53 6.1 Outer Lebesgue Measure . . . . . . . . . . . . . . . . . . . . 53 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7 What Is a Measurable Set? 59 7.1 Identifying Measurable Sets . . . . . . . . . . . . . . . . . . 59 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 8 Decomposition Theorems 69 8.1 Signed Measures And the Hahn Decomposition . . . . . . . 69 8.2 The Radon–Nikody´m Theorem . . . . . . . . . . . . . . . . 73 vii viii Contents 8.3 The Riesz Representation Theorem . . . . . . . . . . . . . . 76 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 9 Creation of Measures 83 9.1 Outer Measure . . . . . . . . . . . . . . . . . . . . . . . . . 83 9.2 Construction of Lebesgue Measure . . . . . . . . . . . . . . 91 9.3 Borel-Stieltjes Measure . . . . . . . . . . . . . . . . . . . . . 92 9.4 Linear Functionals on C(X) . . . . . . . . . . . . . . . . . . 92 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 10 Instances of Measurable Sets 97 10.1 Particular Sets . . . . . . . . . . . . . . . . . . . . . . . . . 97 10.2 Lebesgue Null Sets . . . . . . . . . . . . . . . . . . . . . . . 98 10.3 Invariance under Translation . . . . . . . . . . . . . . . . . 99 10.4 A Lebesgue Measurable Set That Is Not Borel . . . . . . . 100 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 11 Approximation by Open And Closed Sets 103 11.1 Interior Approximation by Closed Sets . . . . . . . . . . . . 105 11.2 Approximation from Inside by Compact Sets . . . . . . . . 106 11.3 Approximation by Intervals . . . . . . . . . . . . . . . . . . 107 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 12 Different Methods of Convergence 109 12.1 Review of Convergence Techniques . . . . . . . . . . . . . . 109 12.2 Convergence in Measure . . . . . . . . . . . . . . . . . . . . 111 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 13 Measure on a Product Space 117 13.1 Product Measures . . . . . . . . . . . . . . . . . . . . . . . 117 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 14 Additivity for Outer Measure 125 14.1 A New Look at Carath´eodory . . . . . . . . . . . . . . . . . 126 14.2 A Few Words about Inner Measure . . . . . . . . . . . . . . 127 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 15 Nonmeasurable Sets And Non-Borel Sets 129 15.1 Nonmeasurable Sets . . . . . . . . . . . . . . . . . . . . . . 129 15.2 Existence of a Measurable Set That Is Not Borel . . . . . . 131 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 16 Applications 135 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Table of Notation 141 Contents ix Glossary 145 Solutions to Selected Exercises 159 References 179 Index 181

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.