Table Of ContentGraduate Texts in Mathematics 262
Editorial Board
S. Axler
K.A. Ribet
For further volumes:
http://www.springer.com/series/136
Daniel W. Stroock
Essentials of Integration
Theory for Analysis
ABC
Daniel W. Stroock
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139
USA
[email protected]
EditorialBoard:
S.Axler K.A.Ribet
SanFranciscoStateUniversity UniversityofCaliforniaatBerkeley
MathematicsDepartment MathematicsDepartment
SanFrancisco,CA94132 Berkeley,CA94720
USA USA
[email protected] [email protected]
ISSN0072-5285
ISBN978-1-4614-1134-5 e-ISBN978-1-4614-1135-2
DOI10.1007/978-1-4614-1135-2
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2011934481
Mathematics Subject Classification (2010): 28-00, 26A42
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Preface
With some justification, measure theory has a bad reputation. It is regarded
by most students as a subject that has little æsthetic appeal and lots of fussy
details. In order to make the subject more palatable, many authors have
chosen to add spice by embedding measure theory inside one of the many
topics in which measure theory plays a central role. In the past, Fourier
analysis was usually the topic chosen, but in recent years Fourier analysis
has been frequently displaced by probability theory. There is a lot to be
said for the idea of introducing a running metaphor with which to motivate
the technical definitions and minutiae with which measure theory is riddled.
However,I1havenotadoptedthispedagogicdevice. Instead,Ihaveattempted
topresentmeasuretheoryasanessentialbranchofanalysis,onethathasmerit
ofitsown. Thus,althoughIoftendigresstodemonstratehowmeasuretheory
answers questions whose origins are in other branches of analysis, this book
is about measure theory, unadorned.
In the first chapter I give a r´esum´e of Riemann’s theory of integration, in-
cludingStieltjes’sextensionofthattheory. MyreasonforincludingRiemann’s
theory is twofold. In the first place, when I turn to Lebesgue’s theory, I want
Riemann’s theory available for comparison purposes. Secondly, and perhaps
more important, I believe that Riemann’s theory provides many of the basic
tools with which one does actual computations. Lebesgue’s theory enables
one to prove equalities between abstract quantities, but evaluation of those
quantities usually requires Riemann’s theory. The final section of Chapter
1 contains an analysis of the rate at which Riemann sums approximate his
integral. In no sense is this section serious numerical analysis. On the other
hand, it gives an amusing introduction to the Euler–Maclaurin formula.
Modern (i.e., apr`es Lebesgue) measure theory is introduced in Chapter
2. I begin by trying to explain why countable additivity is the sine qua
non in Lebesgue’s theory of integration. This explanation is followed by the
derivation of a few elementary properties possessed by countably additive
measures. Inthesecondsectionofthechapter,Idevelopasomewhatprimitive
procedure for constructing measures on metric spaces and then apply this
procedure to the construction of Lebesgue measure λ on RN, the measure
RN
1Contrarytotheconventioninmostmodernmathematicalexpositionandthe
wishes of the GTM editors, I often use the first person singular rather than
the“royalwe”whenIexpectthereadertobeplayingapassiverole. Irestrict
the use of “we” to places, like proofs, where I expect the active participation
of my readers.
v
vi Preface
µ onRdeterminedbyadistributionfunctionF,andtheBernoullimeasures
F
β on {0,1}Z+. Included here are a proof of the way in which Lebesgue
p
measure transforms under linear maps and of the relationship between λ
RN
and β .
1
2
Lebesgue integration theory is taken up in Chapter 3. The basic theory is
covered in the first section, and its miraculous stability (i.e., the Monotone
Convergence and Lebesgue’s Dominated Convergence Theorems as well as
Fatou’s Lemma) is demonstrated in next section. The third section is a bit
of a digression. There I give a proof, based on Riesz’s Sunrise Lemma, of
Lebesgue’s Differentiation Theorem for increasing functions.
The first section of Chapter 4 is devoted to the construction of product
measuresandtheproofofFubini’sTheorem. Asanapplication,inthesecond
section I describe Steiner’s symmetrization procedure and use it to prove
the isodiametric inequality, which I then apply to show that N-dimensional
Hausdorff measure in RN is Lebesgue measure there.
In Chapter 5 I discuss several topics that are tied together by the fact that
they all involve changes of variables. The first of these is the application of
distribution functions to show that Lebesgue integrals can be represented as
Riemann integrals, and the second topic is polar coordinates. Both of these
are in §5.1. The second section contains a proof of Jacobi’s transformation
formula and an application of his formula to the construction of surface mea-
sure for hypersurfaces in RN. My treatment of these is, from a differential
geometric perspective, extremely inelegant: there are no differential forms
here. In particular, my construction of surface measure is concertedly non-
intrinsic. Instead, I have adopted a more geometric measure-theoretic point
of view and constructed surface measure by “differentiating” Lebesgue mea-
sure. Similarly, my derivation in §5.3 of the Divergence Theorem is devoutly
extrinsic and devoid of differential form technology.
Some of the bread and butter inequalities (specifically, Jensen’s, H¨older’s,
andMinkowski’s)ofintegrationtheoryarederivedinthefirstsectionofChap-
ter 6. In the second section, these inequalities are used to study some ele-
mentary geometric facts about the Lebesgue spaces Lp as well as the mixed
LebesguespacesL(p,q). Theresultsobtainedin§6.2areappliedin§6.3tothe
analysis of boundedness properties for transformations defined by kernels on
the Lebesgue space. Particular emphasis is placed on transformations given
byconvolution,forwhichYoung’sinequalityisproved. Thechapterendswith
a brief discussion of Friedrichs mollifiers.
In preparation for Fourier analysis, Chapter 7 begins with a cursory intro-
duction to Hilbert spaces. The basic L2-theory of Fourier series is given in
§7.2andisappliedtheretocompletetheprogram,startedin§1.3ofChapter
1, of understanding the Euler–Maclaurin formula. The elementary theory of
the Fourier transform is developed in §7.3, where I first give the L1-theory
and then the L2-theory. My approach to the latter is via Hermite functions.
Preface vii
The concluding chapter contains several vital topics that were either given
short shrift or entirely neglected earlier. The first of these is the Radon–
Nikodym Theorem, which I, following von Neumann, prove as an application
of Riesz’s Representation Theorem for Hilbert space. The second topic is
Daniell’s theory of integration, which I use first to derive the standard cri-
terion that says when a finite, finitely additive measure admits a countably
additive extension and second to derive the Riesz Representation Theorem
for non-negative linear functionals on continuous functions. The final topic
is Carath´eodory’s method for constructing measures from subadditive func-
tions and its application to the construction of Hausdorff measures on RN.
Although my treatment of Hausdorff measures barely touches on the many
beautiful and deep aspects of this subject, I do show that the restriction of
(N − 1)-dimensional Hausdorff measure to a hypersurface in RN coincides
with the surface measure constructed in §5.2.2.
It is my hope that this book will be useful both as a resource for students
trying to learn measure theory on their own and as text for a course. I have
used it at M.I.T. as the text for a one semester course. However, M.I.T.
students are accustomed to abuse,2 and it is likely that as a text for a one
semester course elsewhere some picking and choosing will be necessary. My
suggestion is that one be sure to cover the first four chapters and the first
sectionsofChapters7and8, perhapsskipping§1.3, §3.3, and§4.2. Depend-
ing on the interests of the students, one can supplement this basic material
with selections from Chapters 5, 6, as well as from material that one skipped
earlier.
There are exercises at the end of each section. Some of these are quite
trivial and others are quite challenging. Especially for those attempting to
learn the subject on their own, I strongly recommend that, at the very least,
myreaderslookatalltheexercisesandsolveenoughofthemtobecomefacile
with the techniques they wish to master. At least for me, it is not possible to
learn mathematics as a spectator.
Daniel W. Stroock
2It has been said that getting an education at M.I.T. is like taking a drink
from a fire hydrant.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Chapter 1 The Classical Theory . . . . . . . . . . . . . . . . 1
1.1 Riemann Integration . . . . . . . . . . . . . . . . . . . . . 1
Exercises for §1.1 . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Riemann–Stieltjes Integration . . . . . . . . . . . . . . . . . 8
1.2.1. Riemann Integrability . . . . . . . . . . . . . . . . . . . 9
1.2.2. Functions of Bounded Variation . . . . . . . . . . . . . . 12
Exercises for §1.2 . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Rate of Convergence . . . . . . . . . . . . . . . . . . . . 20
1.3.1. Periodic Functions . . . . . . . . . . . . . . . . . . . . 20
1.3.2. The Non-Periodic Case . . . . . . . . . . . . . . . . . . 25
Exercises for §1.3 . . . . . . . . . . . . . . . . . . . . . . . 27
Chapter 2 Measures . . . . . . . . . . . . . . . . . . . . . 28
2.1 Some Generalities . . . . . . . . . . . . . . . . . . . . . 28
2.1.1. The Idea . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.2. Measures and Measure Spaces . . . . . . . . . . . . . . . 30
Exercises for §2.1 . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 A Construction of Measures . . . . . . . . . . . . . . . . . 39
2.2.1. A Construction Procedure . . . . . . . . . . . . . . . . 39
2.2.2. Lebesgue Measure on RN . . . . . . . . . . . . . . . . . 45
2.2.3. Distribution Functions and Measures . . . . . . . . . . . 50
2.2.4. Bernoulli Measure . . . . . . . . . . . . . . . . . . . . 51
2.2.5. Bernoulli and Lebesgue Measures . . . . . . . . . . . . . 55
Exercises for §2.2 . . . . . . . . . . . . . . . . . . . . . . . 57
Chapter 3 Lebesgue Integration . . . . . . . . . . . . . . . 62
3.1 The Lebesgue Integral . . . . . . . . . . . . . . . . . . . 62
3.1.1. Some Miscellaneous Preliminaries . . . . . . . . . . . . . 62
3.1.2. The Space L1(µ;R) . . . . . . . . . . . . . . . . . . . 70
Exercises for §3.1 . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Convergence of Integrals . . . . . . . . . . . . . . . . . . 73
3.2.1. The Big Three Convergence Results . . . . . . . . . . . . 74
3.2.2. Convergence in Measure . . . . . . . . . . . . . . . . . 77
3.2.3. Elementary Properties of L1(µ;R) . . . . . . . . . . . . . 82
ix
x Contents
Exercises for §3.2 . . . . . . . . . . . . . . . . . . . . . . . 84
3.3 Lebesgue’s Differentiation Theorem . . . . . . . . . . . . . 87
3.3.1. The Sunrise Lemma . . . . . . . . . . . . . . . . . . . 87
3.3.2. The Absolutely Continuous Case . . . . . . . . . . . . . 89
3.3.3. The General Case . . . . . . . . . . . . . . . . . . . . 94
Exercises for §3.3 . . . . . . . . . . . . . . . . . . . . . . . 96
Chapter 4 Products of Measures . . . . . . . . . . . . . . . 100
4.1 Fubini’s Theorem . . . . . . . . . . . . . . . . . . . . . 100
Exercises for §4.1 . . . . . . . . . . . . . . . . . . . . . . . 104
4.2 Steiner Symmetrization . . . . . . . . . . . . . . . . . . . 106
4.2.1. The Isodiametric inequality . . . . . . . . . . . . . . . . 106
4.2.2. Hausdorff’s Description of Lebesgue’s Measure . . . . . . . 108
Exercises for §4.2 . . . . . . . . . . . . . . . . . . . . . . . 112
Chapter 5 Changes of Variable . . . . . . . . . . . . . . . . 113
5.1 Riemann vs. Lebesgue, Distributions, and Polar Coordinates . . 113
5.1.1. Riemann vs. Lebesgue . . . . . . . . . . . . . . . . . . 113
5.1.2. Polar Coordinates . . . . . . . . . . . . . . . . . . . . 117
Exercises for §5.1 . . . . . . . . . . . . . . . . . . . . . . . 119
5.2 Jacobi’s Transformation and Surface Measure . . . . . . . . . 121
5.2.1. Jacobi’s Transformation Formula . . . . . . . . . . . . . 121
5.2.2. Surface Measure . . . . . . . . . . . . . . . . . . . . . 124
Exercises for §5.2 . . . . . . . . . . . . . . . . . . . . . . . 132
5.3 The Divergence Theorem . . . . . . . . . . . . . . . . . . 137
5.3.1. Flows Generated by Vector Fields . . . . . . . . . . . . . 137
5.3.2. Mass Transport . . . . . . . . . . . . . . . . . . . . . 139
Exercises for §5.3 . . . . . . . . . . . . . . . . . . . . . . . 142
Chapter 6 Basic Inequalities and Lebesgue Spaces . . . . . . 146
6.1 Jensen, Minkowski, and H¨older . . . . . . . . . . . . . . . 146
Exercises for §6.1 . . . . . . . . . . . . . . . . . . . . . . . 151
6.2 The Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . 153
6.2.1. The Lp-Spaces . . . . . . . . . . . . . . . . . . . . . 153
6.2.2. Mixed Lebesgue Spaces . . . . . . . . . . . . . . . . . 158
Exercises for §6.2 . . . . . . . . . . . . . . . . . . . . . . . 161
6.3 Some Elementary Transformations on Lebesgue Spaces . . . . . 163
6.3.1. A General Estimate for Linear Transformations . . . . . . . 163
6.3.2. Convolutions and Young’s inequality . . . . . . . . . . . . 165
6.3.3. Friedrichs Mollifiers . . . . . . . . . . . . . . . . . . . 168
Exercises for §6.3 . . . . . . . . . . . . . . . . . . . . . . . 171
