REAL ANALYSIS

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REAL ANALYSISIbookroot October 20, 2007PrincetonLecturesinAnalysisI Fourier ANALYSIS : An IntroductionII Complex AnalysisIII Real ANALYSIS : measure Theory, Integration, andHilbert SpacesIV Functional ANALYSIS : Introductionto Further Topics in AnalysisPrinceton Lectures in AnalysisIIIREAL ANALYSISMeasure Theory, Integration, andHilbert SpacesElias M. Stein&Rami ShakarchiPRINCETON UNIVERSITY PRESSPRINCETON AND OXFORDCopyright 2005 by Princeton University PressPublished by Princeton University Press, 41 William Street,Princeton, New Jersey 08540In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TWAll Rights ReservedLibrary of Congress Control Number 2004114065ISBN 978-0-691-11386-9British Library Cataloging-in-Publication Data is availableThe publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printedPrinted on acid-free paper. in the United States of America5 7 9 10 8 643To my grandchildrenCarolyn, Alison, my parentsMohamed & Mireilleand my in the spring of 2000, a series of four one-semester courseswere taught at Princeton University whose purpose was to present, inan integrated manner, the core areas of ANALYSIS .

Chapter 6. Abstract Measure and Integration Theory 262 1 Abstract measure spaces 263 1.1 Exterior measures and Carath¶eodory’s theorem 264 1.2 Metric exterior measures 266 1.3 The extension theorem 270 2 Integration on a measure space 273 3 Examples 276 3.1 Product measures and a general Fubini theorem 276

  Measure, Space, Measure spaces