An Introductory Course in Lebesgue Spaces
This book is devoted exclusively to Lebesgue spaces and their direct derived spaces. Unique in its sole dedication, this book explores Lebesgue spaces, distribution functions and nonincreasing rearrangement. Moreover, it also deals with weak, Lorentz and
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René Erlín Castillo Humberto Rafeiro
Canadian Mathematical Society Société mathématique du Canada
An Introductory Course in Lebesgue Spaces
Canadian Mathematical Society Soci´et´e math´ematique du Canada Editors-in-Chief R´edacteurs-en-chef K. Dilcher K. Taylor Advisory Board Comit´e consultatif M. Barlow H. Bauschke L. Edelstein-Keshet N. Kamran M. Kotchetov
More information about this series at http://www.springer.com/series/4318
Ren´e Erl´ın Castillo
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Humberto Rafeiro
An Introductory Course in Lebesgue Spaces
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Ren´e Erl´ın Castillo Universidad Nacional de Colombia Bogot´a, Colombia
Humberto Rafeiro Pontificia Universidad Javeriana Bogot´a, Colombia
ISSN 1613-5237 ISSN 2197-4152 (electronic) CMS Books in Mathematics ISBN 978-3-319-30032-0 ISBN 978-3-319-30034-4 (eBook) DOI 10.1007/978-3-319-30034-4 Library of Congress Control Number: 2016935410 Mathematics Subject Classification (2010): 46E30, 26A51, 42B20, 42B25 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland
A la memoria de Julieta mi mam´a-abuela A mi esposa Hilcia del Carmen A mis hijos: Ren´e Jos´e Manuel Alejandro Irene Gabriela y Renzo Rafael R.E.C.
a` Daniela pelo seu amor e paciˆencia H.R.
Preface
This book is not a treatise on Lebesgue spaces, since this would not be a feasible work due to the extension of their usage, e.g., in physics, probability, statistics, economy, engineering, among others. The objective is more realistic, being the introduction of the reader to the study of different variants of Lebesgue spaces and the common techniques used in this area. Since the Lebesgue spaces measure integrability of a function, they can be seen as the father of all integrable function spaces where more fine properties of functions are sought. We can find many books where the subject of L
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