Measure, Integral, Derivative: A Course on Lebesgue's Theory

154 Pages·2013·0.975 MB·English

by  Sergei Ovchinnikov (auth.)

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Sergei Ovchinnikov
Measure, Integral, Derivative
A Course on Lebesgue’s Theory
123
SergeiOvchinnikov
DepartmentofMathematics
SanFranciscoStateUniversity
SanFrancisco,CA,USA
ISSN0172-5939 ISSN2191-6675(electronic)
ISBN978-1-4614-7195-0 ISBN978-1-4614-7196-7(eBook)
DOI10.1007/978-1-4614-7196-7
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To the memory of my best
and truest friend, Timosha

Preface
This book originated from the notes I kept while teaching the graduate
course Analysis on Lebesgue’s theory of integration and differentiation at
San Francisco State University (SFSU). This course was rightly considered
by many students as a difficult one, mainly because some ideas and proofs
were presented in their textbooks in unnatural and counterintuitive ways,
albeit rigorous ones. These students also had problems connecting the mate-
rial they learned in undergraduate real analysis classes with this course. The
Mathematics Department of SFSU is a master’s mathematics department.
Most students who receiveda MS degree from our department do not pursue
ahigherdegree.WhenteachingthiscourseIwantedmystudentstogetafeel
for the theory, appreciate its importance, and be ready to learn more about
it, should the need arise. Consequently, my goal in writing this book was to
present Lebesgue’s theory in the most elementary way possible by sacrificing
the generality of the theory. For this, the theory is built constructively for
measures and integrals over bounded sets only. However, the reader will find
allmaintheoremsofthetheoryhere,ofcoursenotintheirultimategenerality.
The first chapter presents selected topics from the real analysis that I
felt are needed to review in order to fill the gaps between what the reader
probably learned some time ago or missed completely and what is required
to master the material presented in the rest of the book. For instance, one
can hardly find properties of summable families (Sect.1.4) in textbooks on
real analysis. Several conventions that are used throughout the book are also
found in Chap.1.
The Lebesgue measure of a bounded set and measurable functions are
the subject of the second chapter. Because bounded open and closed sets
haverelativelysimplestructures,theirmeasuresareintroducedfirst.Thenthe
outer and inner measures of a bounded set are introduced by approximating
the set by open and closed sets, respectively. A measurable set is defined as
a bounded set for which its inner and outer measures are equal; its Lebesgue
measure is the common value of these two measures. We proceed then by
VII
VIII Preface
establishingstandardpropertiesoftheLebesguemeasureandmeasurablesets.
Lebesgue measurable functions and their convergence properties are covered
in the last two sections of Chap.2. Undoubtedly, the highest point of this
chapter is Egorov’s Theorem, which is important in establishing convergence
properties of integrals in Chap.3.
I follow most expositions in Chap.3 where main elements of the theory
of Lebesgue integral are presented. Again the theory is developed for func-
tions over bounded sets only. However, the main convergence theorems—the
Bounded Convergence Theorem, the Monotone Convergence Theorem, and
Dominated Convergence Theorem—are proved in this chapter, establishing
the “passage of the limit under the integral sign.”
The main topics ofChap.4 areLebesgue’s theoremabout differentiability
of monotone functions and his versions of the fundamental theorems of cal-
culus. I chose to present functions of bounded variations (BV-functions) and
their properties first and then prove the Lebesgue theorem for BV-functions.
The proof is elementary albeit a nontrivial one. To make it more accessible,
I dissect the proof into a number of lemmas and two theorems. The last two
sectionsofChap.4coverabsolutelycontinuousfunctionsandthefundamental
theorems of calculus due to Lebesgue.
A distinguished feature of this book is that it limits attention in Chaps.2
and 3 to bounded subsets of the real line. In the Appendix, I present a way
to remedy this limitation.
Thereare187exercisesinthe book(thereisanexercisesectionattheend
of each chapter). Most exercises are “proof” problems, that is, the reader is
invited to prove a statement in the exercise.
I have received help from many people in the process of working on the
drafts of this book. First and foremost, I am greatly indebted to my students
for correcting several errors in the lecture notes from which this text was de-
rived and providing other valuable feedback. I wish to thank my colleague
Eric Hayashi and an anonymous referee for reading parts of the manuscript
carefullyandsuggestingmanymathematicalandstylisticcorrections.Myspe-
cial thanks goto Sheldon Axler for his endorsementofthis projectand many
comments which materially improvedthe original draftof the book. Lastbut
not least, I wish to thank my Springer editor Kaitlin Leach for her support
throughout the preparationof this book.
Berkeley, CA, USA Sergei Ovchinnikov
Contents
1 Preliminaries .............................................. 1
1.1 Sets and Functions ...................................... 1
1.2 Sets and Sequences of Real Numbers....................... 4
1.3 Open and Closed Sets of Real Numbers .................... 9
1.4 Summation on Infinite Sets ............................... 14
Notes....................................................... 19
Exercises ................................................... 20
2 Lebesgue Measure ......................................... 27
2.1 The Measure of a Bounded Open Set ...................... 28
2.2 The Measure of a Bounded Closed Set ..................... 33
2.3 Inner and Outer Measures ................................ 38
2.4 Measurable Sets......................................... 42
2.5 Translation Invariance of Measure ......................... 48
2.6 The Class of Measurable Sets ............................. 50
2.7 Lebesgue Measurable Functions ........................... 52
2.8 Sequences of Measurable Functions ........................ 55
Notes....................................................... 59
Exercises ................................................... 60
3 Lebesgue Integration ...................................... 65
3.1 Integration of Bounded Measurable Functions............... 65
3.2 Properties of the Integral................................. 68
3.3 Convergence ............................................ 76
3.4 Integration of Nonnegative Measurable Functions............ 78
3.5 General Lebesgue Integral ................................ 84
3.6 Comparison of Riemann and Lebesgue Integrals ............. 88
Notes....................................................... 90
Exercises ................................................... 91
IX