A Course in Analysis: Vol. III: Measure and Integration Theory, Complex-Valued Functions of a Complex Variable

784 Pages·2017·5.71 MB·English

by  NIELS JACOB, KRISTIAN P EVANS

A Course in Analysis (New)
Vol. I
Part 1 Introductory Calculus
Part 2 Analysis of Functions of One Real Variable
Vol. II
Part 3 Differentiation of Functions of Several Variables
Part 4 Integration of Functions of Several Variables
Part 5 Vector Calculus
Vol. III
Part 6 Measure and Integration Theory
Part 7 Complex-valued Functions of a Complex Variable
Vol. IV
Part 8 Fourier Analysis
Part 9 Ordinary Differential Equations
Part 10 Introduction to the Calculus of Variations
Vol. V
Part 11 Functional Analysis
Part 12 Operator Theory
Part 13 Theory of Distributions
Vol. VI
Part 14 Partial Differential Equations: Classical Theory
Part 15 Partial Differential Equations and Distributions
Vol. VII
Part 16 Differential Geometry of Curves and Surfaces
Part 17 Differentiable Manifolds
Part 18 Lie Groups

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Library of Congress Cataloging-in-Publication Data
Jacob, Niels.
A course in analysis / by Niels Jacob (Swansea University, UK),
Kristian P. Evans (Swansea University, UK).
volumes cm
Includes bibliographical references and index.
Contents: volume 1. Introductory calculus, analysis of functions of one real variable
ISBN 978-9814689083 (hardcover : alk. paper) -- ISBN 978-9814689090 (pbk : alk. paper)
1. Mathematical analysis--Textbooks. 2. Mathematics--Study and teaching (Higher)
3. Calculus--Textbooks. I. Evans, Kristian P. II. Title.
QA300.J27 2015
515--dc23
2015029065
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or
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ISBN 978-981-3221-59-8
Printed in Singapore
Preface
A detailed description of the content of Volume III of our Course in Analysis
will be provided in the introduction. Here we would like to take the oppor-
tunity to thank those who have supported us in writing this volume. We
owe a debt of gratitude to James Harris who has typewritten the majority
of the manuscript. Thanks for typewriting further parts are expressed to
Saroj Limbu and James Morgan. Huw Fry, James Harris and Elian Rhind
undertook a lot of proofreading for which we are grateful. We also want to
thank the Department of Mathematics, the College of Science and Swansea
University for providing us with funding for typewriting.
It turned out that R. Schilling was working on the second edition of his book
“Measure, Integration and Matingales” [75] while we were working on Part
6. This led to many interesting discussions between Dresden and Swansea
from which we could benefit a lot and for which we are grateful.
Finally we want to thank our publisher, in particular Tan Rok Ting and Ng
Qi Wen, for a pleasant collaboration.
Niels Jacob
Kristian P. Evans
Swansea, January 2017
v
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Introduction
Thethirdvolumeofour“CourseinAnalysis”coverstwotopicsindispensable
for every mathematician whether specializing in analysis or not. Part 6 dis-
cusses Lebesgue’s theory of integration and first consequences in the theory
of a real-valued function of a real variable. Part 7 introduces the theory of
complex-valued functions of one complex variable - traditionally just called
the “the theory of functions”.
The advanced theory of real-valued functions, Fourier analysis, functional
analysis, the theory of dynamical systems, partial differential equations or
the calculus of variations are all topics in analysis depending on Lebesgue’s
integration theory. But in addition, probability theory, parts of geometry
(fractal sets) and more applied subjects such as information theory or opti-
mization needs a proper understanding of Lebesgue integration. Moreover,
many mathematicians will agree that the structure of the real line is one of
the most complicated structures, if not the most complicated structure of
all, we have to deal with. Many problems leading to a better understanding
of R and in fact leading to deep developments in the foundation of mathe-
matics (mathematical logic and set theory) we encounter when relating the
topologicalstructure ofR asinduced by theEuclidean metric to theproblem
of defining and determining the size of subsets of R. Often the underlying
model of set theory determines the relation between topology and measure
theory. In our treatise we always assume ZF as underlying model of set the-
ory but we do not spend much time on investigating the problems mentioned
above, they are the topic of different and more advanced courses.
WewilldescribethecontentofPart6inmoredetailbelow,howeverwewould
like to mention the influence of [11] in our presentation. The forerunner of
H. Bauer’s monograph [11], i.e. [10], was the standard text book on measure
theory in Germany for many decades. When [10] was split into two books
the first named author was heavily involved in proof reading and discussing
the material.
The theoryof functionsis key toevery other advanced theory in analysis and
at the same time it is needed as a tool in many other applied mathematical
disciplines such as mechanics or in fields such as electrical engineering or
physics. But much more holds: holomorphic functions, i.e. complex differen-
vii
A COURSE IN ANALYSIS
tiable complex-valued functions of a complex variable enter into many fields
in pure mathematics such as number theory, algebraic geometry, representa-
tiontheory, differentialgeometryorcombinatoricsandmanyothers. Itisfair
to say that without a proper knowledge of function theory no undergraduate
education in pure or applied mathematics can be viewed as being complete.
A more detailed discussion of Part 7 follows below.
Before going into the details we have to add two remarks. Firstly, originally
we also planned to include in our third volume Part 8: Fourier analysis.
While writing Part 6 and 7 it emerged that a better strategy is to add more
(advanced or specialized) material from the two theories covered in Part 6
and7alreadyhereandnotinlaterpartswheretheywillbeneeded, e.g. diffe-
rentiability propertiesofreal-valued functions, Sard’stheorem, densesubsets
in Lp-spaces and the Friedrichs mollifier, or hypergeometric function, elliptic
integrals and elliptic functions, just to mention a few. This will of course
lead to some alteration of the previous plan of arranging the entire material.
Secondly, thereader will notice adifferent modeofreferring totheliterature.
We now sometimes deal with topics which admit quite different approaches
and representations. Clearly we are influenced by authors who dealt with
this material before and of course we want to and we have to give fair credit
where appropriate. The fact thatwe have meanwhile reached more advanced
material is also taken into account when in some but not many proofs we
leave straightforward calculations to the reader, something we have strictly
avoided in the first two volumes. Some of our problems are now more invol-
ved and some of the solutions are more brief, again a reflection of the fact
that we now address more (mathematically) matured readers.
InChapter1weintroduceσ-fields, theirgeneratorsandmeasures, andChap-
ter 2 is devoted to the Carath´eodory extension theorem. A discussion of the
Lebesgue-Borel measure and of the Hausdorff measure as well as the Haus-
dorff dimension follows in Chapter 3. In particular the Cantor set is treated
in great detail. Measurable mappings are the topic of Chapter 4. The stan-
dard approach to define the Lebesgue integral with respect to a measure is
developed in Chapter 5, and Chapter 6 starts to handle measures with densi-
ties. We prove the Radon-Nikodym theorem as we prove the transformation
theorem for Lebesgue integrals. The role of sets of measure zero and almost
everywhere statements are treated in Chapter 7 along with the main conver-
gence results, especially the dominated convergence theorem. We also look
viii
INTRODUCTION
at spaces of p-fold integrable functions. These results are then applied in
Chapter 8 to prove typical theorems about the interchanging of limits such
as the continuity or the differentiability of parameter dependent integrals.
We prove Jensen’s inequality and discuss the relation between the Lebesgue
integral and the Riemann integral. By this we fill in some of the gaps left
by dealing with the Riemann integral in higher dimensions in Volume II.
This discussion includes improper integrals in particular as well as the intro-
duction of the Lp-spaces. Product integrations and most of all the theorems
of Tonelli and Fubini are the main content of Chapter 9. As an important
application we look at integration with respect to the distribution function
and we give some examples of theoretical interest. We also provide a com-
plete proof of Minkowski’s integral inequality. From our point of view the
content of Chapters 1-9 forms the core of any module on measure and in-
tegration theory suitable for the purposes of analysis as well as probability
theory. Chapters 10-12 deal with topics which are interesting by themselves
but they are also major tools in areas such as Fourier analysis, functional
analysis or (partial) differential equations. Chapter 10 is devoted to the con-
volution of functions and measures and we prove the density of continuous
functions with compact support in the spaces Lp(Rn), 1 p < . Further
≤ ∞
we handle the Friedrichs mollifier which will turn out to become a first class
tool in many later considerations. Finally, we have a first look at convolu-
tion operators. Lebesgue’s theory of differentiation is our topic in Chapter
11. After having proved the Vitali covering theorem we introduce absolu-
tely continuous functions and functions of bounded variation and study their
relations. The key results are a new version of the fundamental theorem
of calculus and Lebesgue’s differentiation theorem, the proof of which is gi-
ven by making use of the Hardy-Littlewood maximal function. Eventually
in Chapter 12 we discuss three special results which will become important
later: a version of Sard’s theorem on the measure of the critical points of a
differentiablemapping,Lusin’stheoremwhichisfollowedbyafirstdiscussion
of weak convergence and the Kolmogorov-Riesz theorem which characterises
relative compact sets in Lp-spaces.
Ourtreatmentofcomplex-valuedfunctionsofacomplexvariablestartswitha
brief recollection of the complex numbers including convergence of sequences
and series. In Chapter 14 we embark onto a small digression and summa-
rize obvious properties of complex-valued functions defined on an arbitrary
set and Chapter 15 is devoted to the geometry of the plane and complex
ix