Differentiation of Integrals in Rn
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Series: Universidad Complutense de Madrid Advisers: A Dou and M. de Guzman
481 Miguel de Guzman
Differentiation of Integrals in Rn
Sprinqer-Verlaq
Berlin· Heidelberg· NewYork 1975
Author Prof. Miguel de Guzman Facultad de Matematicas Universidad Complutense de Madrid Madrid 3/Spain
Library of Congress Cataloging in Publication Data
Guzman, Miguel de, 1936Differentiation of integrals in an (Lecture notes in mathematics ; 481) Bibliography: p. Includes index. 1. Integrals, Generalized. 2. Measure theory. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 481. QA3.L28 no. 4 81 [QA312J 510'.8s [515'.43J 7525635
AMS Subject Classifications (1970): 26A24, 28A15 ISBN 3-540-07399-X Springer-Verlag Berlin· Heidelberg· New York ISBN 0-387-07399-X Springer-Verlag New York· Heidelberg' Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, "broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin· Heidelberg 1975 Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
DEDICATED TO MAYTE Migue1
2
and Mayte
2
PREFACE The work presented here deals with the local aspect of the differentiation theory of integrals. This theory takes its origin in thewellknown theorem of Lebesgue [1910]: Let f be a real function in L1(Rn). Then, for almost every x e Rn we have, for every sequence of open Euclidean balls B(x,r
lim (l/IB(x,r k ) ! )
J
B(x,r
k)
centered at x such that r
f(y)dy = f(x)
as
k
0,
k
k)
One could think that the fact that one takes here the limit of the means over Euclidean balls instead of taking them over other type of sets contracting to the point x might well be irrelevant. It was not until about 1927 that H. Bohr exhibited an example, first published by Caratheodory [1927], showing that intervals in R2 (i.e. rectangles with sides parallel to the axes) behave much worse than cubic intervals or circles with regard to a covering property (Vitali's lemma) that was fundamental for the result of Lebesgue. So it became a challenging problem to find out whether the replacement of Euclidean balls by intervals centered at the point x in the Lebesgue theorem would lead to a true statement or not. The first result in this direction was the so-called strong density theorem, first proved by Saks [1933], stating that if the function f is the characteristic function of a measurable set, then Ecuclidean balls can be replaced by intervals. Later on Zygmund [1934] showed that this can also be done if f is in any space LP(Rn), with 1 < P ,
and a year later Jessen, Marcin-
kiewicz and Zygmund [1935] proved that the same is valid if f is in L(l+log+ L)n-1(Rn). On the other hand Saks [1934] proved tha
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