Yeh - Problems and Proofs in Real Analysis

Intended as a self-study volume. ISBN 978-981-4578-50-9 (softcover: alk. paper) 1. Mathematical analysis--Study and teaching. 2. Yeh, J. Yeh, J.

Real analysis. ll. Title. QA312.Y445 2014 515'.8--dc23 2013041974 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2014 by World Scientific Publishing Co. Ltd. All rights reserved. All rights reserved.

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Betty ContentsPreface This volume consists of proofs of the problems in the monograph Real Analysis: Theory of Measure and Integration, 3rd Edition. Alternate proofs are included when appropriate to show different approaches to the problem or different techniques in constructing a proof. J. J.

Yeh Corona del Mar, California September, 2013 1 Measure on a -algebra of SetsProb.1.1. Given two sequences of subsets (En : n N ) and (Fn : n N ) of a set X. (a) Show that (b) Show that (c) Show that and exist, then and exist and moreover Proof. Let (An : n N ) be a sequence of subsets of X. According to Lemma 1.7, consists of every xX such that xAn for all but finitely many n N and consists of every xX such that xAn for infinitely many n N . This is the basis for the proof of the chain of inclusions. 1.1. Let us prove . 1.1. Let us prove .

Let . Then we have . If then xEn for all but finitely many n N and then xEnFn for all but finitely many n N and therefore . Similarly if then . This proves . 1.2. Let us prove .

Let . Then xEnFn for all but finitely many n N . Suppose xFn for infinitely many n N . Then . On the other hand if xFn for only finitely many n N , then since for all but finitely many n N , we must have xEn for all but finitely many n N and then . This shows that if then so that .

This proves . 1.3. Let us prove . Since , we have . Since EnEnFn for every n N , we have . Similarly . Therefore