A Concise Introduction to Measure Theory
This undergraduate textbook offers a self-contained and concise introduction to measure theory and integration. The author takes an approach to integration based on the notion of distribution. This approach relies on deeper properties of the Riemann integ
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A Concise Introduction to Measure Theory
A Concise Introduction to Measure Theory
Satish Shirali
A Concise Introduction to Measure Theory
123
Satish Shirali Formerly of Panjab University Chandigarh, India and The University of Bahrain Zallaq, Bahrain
ISBN 978-3-030-03240-1 ISBN 978-3-030-03241-8 https://doi.org/10.1007/978-3-030-03241-8
(eBook)
Library of Congress Control Number: 2018960220 Mathematics Subject Classification (2010): 28-01, 28A05, 28A10, 28A12, 28A20, 28A25, 28A35 © Springer Nature Switzerland AG 2018 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. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
The concept of measure in an abstract space, and of the integral with respect to it, has played a fundamental role in analysis and probability in the decades after the pathbreaking work of H. Lebesgue in 1902. It has therefore been an essential component of a mathematics curriculum, accessible to any student who has acquired a facility with basic analysis, including the Heine–Borel theorem, the theory of Riemann integration, infinite series, and also with the use of sets and quantifiers (“for all” and “there exists”) in definitions and proofs. This book assumes such a facility on the part of the reader and also an understanding of the notions of countability and Cartesian product. Equivalence classes and the axiom of choice are needed only once. It is well known that the abstract integral can be interpreted as an improper Riemann-type integral of a related function on the real line, thereby rendering it relatively more concrete. The related function is one that closely resembles what is called the cumulative distribution in probability and statistics and may reasonably be called simply the distribution. This book introdu
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