Real Analysis: Measures, Integrals and Applications

780 Pages·2013·5.63 MB·English

by  Boris Makarov

Universitext
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Boris Makarov (cid:2) Anatolii Podkorytov
Real Analysis:
Measures,
Integrals and
Applications
BorisMakarov AnatoliiPodkorytov
MathematicsandMechanicsFaculty MathematicsandMechanicsFaculty
StPetersburgStateUniversity StPetersburgStateUniversity
StPetersburg,Russia StPetersburg,Russia
TranslatedfromtheRussianlanguageedition:
Lekcii po vewestvennomu analizu
byB.M. Makarov, A.N. Podkorytov(B.M.Makarov,A.N.Podkorytov)
Copyright©BHV-Peterburg(BHV-Petersburg)2011
Allrightsreserved.
ISSN0172-5939 ISSN2191-6675(electronic)
Universitext
ISBN978-1-4471-5121-0 ISBN978-1-4471-5122-7(eBook)
DOI10.1007/978-1-4471-5122-7
SpringerLondonHeidelbergNewYorkDordrecht
LibraryofCongressControlNumber:2013940613
Mathematics Subject Classification: 28A12, 28A20, 28A25, 28A35, 28A75, 28A78, 28B05, 31B05,
42A20,42B05,42B10
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Diagram of Chapter Dependency
v
Preface to the English Translation
This book reflects our experience in teaching at the Department of Mathematics
and Mechanics of St. Petersburg State University. It is aimed primarily at readers
makingtheirfirstacquaintancewiththesubject.
Lecturecoursesonmeasuretheoryandintegrationareoftenconfinedtoabstract
measuretheory,withlittleattentionpaid tosuch topicsas integrationwith respect
to Lebesgue measure, its transformation under a diffeomorphism and so on—that
is, topics that are more special but no less important for applications. Believing
thatsuchreticenceiscounterproductive,wechooseanapproachthatavoidsitand
combinesgeneralnotionswithclassicalspecialcases.
A substantial part of the book is devoted to examples illustrating the obtained
resultsbothinandbeyondtheframeworkofmathematicalanalysis,inparticular,in
geometry.Theexercisesappearingattheendofalmosteverysectionservethesame
purpose.
IntheEnglishtranslationweusethree-digitnumbersforsections.Thefirstdigit
refers to a Chapter, the second to a Section within the Chapter and the third to
Subsection. When referencing to a statement we give the number of a Subsection
whichcontainsit.E.g.,Lemma7.5.4wouldmeanalemmafromSect.7.5.4.
ComparingwithRussianedition,wehaveextendedthebookbyadding,inpar-
ticular,thenewSects.6.1.3and6.2.6.
Taking into account the difference between curricula in Russia and the West,
as well as the considerable volume of our book, we think it necessary to say sev-
eral words about how to use it, and we draw the reader’s attention to the chapter
dependencychart.Areaderinterestedonlyinanintroductiontothefoundationsof
measuretheoryandintegrationmayprefertoreadonlythosesectionsofChaps.1–5
thatarenotmarkedwitha(cid:2).Thissymbolindicatessectionsthatcontaineithersome
illustrativematerial(e.g.,Sects.2.8,6.6–6.7,7.2–7.3,8.7,10.2,10.6),orsomeop-
tional information that can be omitted in the first reading (e.g., Sects. 1.6, 4.11,
5.5–5.6, 6.5, 7.4, 8.8, 10.4, 12.1–12.3), or else material used outside Chaps. 1–5
(Sects.2.6,3.4,4.9).ThematerialofSects.1.1–1.4,2.1–2.5,3.1–3.2,4.1–4.8,5.1–
5.4canbetakenasabasisforatwosemestercourseonthefoundationsofmeasure
vii
viii PrefacetotheEnglishTranslation
theory and integration. Time permitting, the course can be extended by including
thematerialofSects.6.1–6.2,3.3,4.9,6.4.
Thebookcanalsobeusedforcoursesaimedatstudentsfamiliarwiththenotion
ofintegrationwithrespecttoameasure.Thereisasufficientlywidechoiceofsuch
coursesdevotedtorelativelynarrowtopicsofrealanalysis.Forexample:
• Themaximalfunctionanddifferentiationofmeasures(Sects.2.7,4.9,11.2,11.3).
• Surfaceintegrals(Sects.2.6,8.1–8.6).
• Functionalsinspacesofmeasurableandcontinuousfunctions(Sects.11.1–11.2,
Chap.12).
• Approximateidentitiesandtheirapplications(Sects.7.5–7.6,Chap.9).
• FourierseriesandtheFouriertransform(Chaps.9,10).
• A course covering only the preliminaries of the theory of Fourier series and
theFouriertransformmaybebased,forexample,onSects.9.1.1–9.1.3,10.1.1–
10.1.4,10.3.1–10.3.6,10.5.1–10.5.4.
Acknowledgments We are deeply indebted to Springer for publishing our book
andwearehappytoseeitreachamuchwideraudienceviaitsEnglishtranslation.
WearegratefultoV.P.Havinwhoattractedthepublisher’sattentiontotheRussian
editionofourbooksoonafteritspublication.
In the process of preparing and publishing the volume, we have been helped
by several people to whom we would like to express our thanks. In particular,
the anonymous referees provided us with constructive comments and suggestions.
B.M.Bekker,A.A.Lodkin,F.L.NazarovandN.V.Tsilevichundertookthehardtask
oftranslation.WewerealsomostfortunatetoreceivefeedbackfromA.I.Nazarov,
F.L. Nazarov and O.L. Vinogradov, which helped us to improve the text in many
places. Our special thanks go to Joerg Sixt, the Springer Editor, for his invaluable
helpandencouragementinpreparingthismanuscript.
Wealsowishtoacknowledgethehelpandsupportwereceivedfromourfriends,
families and colleagues who read and commented upon various drafts and con-
tributedtothetranslation.
Our special thanks go to O.B. Makarova (who happened to be a granddaugh-
ter of the first of co-authors) who very competently and patiently conducted our
correspondencerelatedtopublicationofthisbookandhelpedusenormouslywith
proofreading.
The translation was carried out with the support of the St. Petersburg State
University program “Function theory, operator theory and their applications”
6.38.78.2011.
StPetersburg,Russia BorisMakarov
AnatoliiPodkorytov
Preface
Measure theory has been an integral part of undergraduate and graduate curricula
in mathematics for a long time now. A number of texts in this subject area have
becomewell-establishedandwidelyused.Forexample,onemightrecallbooksby
B.Z.Vulih[Vu],A.N.KolmogorovandS.V.Fomin[KF],nottomentiontheclassi-
calmonographbyP.Halmos[H].However,booksonmeasuretheorytypicallytreat
itasanisolatedsubject,whichmakesitdifficulttoincludeitinageneralcoursein
analysisinanaturalandseamlessway.Forexample,theinvarianceoftheLebesgue
measureiseitheromittedentirely,orconsideredasaspecialcaseoftheinvariance
oftheHaarmeasure.Quiteoften,thequestionofhowLebesguemeasuretransforms
underdiffeomorphismsisleftout.Ontheotherhand,mostintroductorycourseson
integrationarestillbasedonthetheoryoftheRiemannintegral.Asaresult,thestu-
dentsareforcedtoabsorbnumerous,howeversimilar,definitionsbasedonRiemann
sumscorrespondingtovarioussituations,suchasdoubleintegrals,tripleintegrals,
line integrals, surface integrals and so on. They must also overcome the unneces-
sarytechnicalcomplicationscausedbythelackofasufficientlygeneralapproach.
Typical examples of such difficulties include justifying the change of the order of
integrationandtakinglimitsundertheintegralsign.
Forthisreason,oneoftenfacesatwo-tierexpositionofthetheoryofintegration,
where at the first stage the notion of measure is not discussed at all, and later the
elementary topics are never revisited, leaving the task of reconciling the various
approachestothestudent.Theauthorsaimtoeliminatethisdivideandprovidean
exposition of the theory of the integral that is modern, yet easily integrated into a
generalcourseinanalysis.Thisencapsulatesinatextbooktheestablishedpracticeat
theDepartmentofMathematicsandMechanicsoftheUniversityofSt.Petersburg.
Thispracticeisbasedonanideaintroducedintheearly60sbyG.P.Akilovandfirst
implementedbyV.P.Havinduringtheacademicyear1963–1964.
The main emphasis of the book is on the exposition of the properties of the
Lebesgueintegralanditsvariousapplications.Thisapproachdeterminedthestyle
ofexpositionaswellasthechoiceofthematerial.Itisourhopethatthereaderwho
mastersthefirstthirdofthebookwillbesufficientlypreparedtostudyanyareaof
ix
x Preface
mathematicsthatreliesuponthegeneraltheoryofmeasure,suchas,amongothers,
probabilitytheory,functionalanalysisandmathematicalphysics.
Applicationsofthetheoryofintegrationconstituteasubstantialpartofthisbook.
In addition to some elements of harmonic analysis, they also include geometric
applications, among which the reader will find both classical inequalities, such as
theBrunn–Minkowskiandisoperimetricinequalities,andmorerecentresults,such
astheproofofBrouwer’stheoremonvectorfieldsonthespherebasedonachange
ofvariables,theK.Ballinequalityandothers.Inordertoillustratetheeffectiveness
and applicability of the theorems presented, and to give the reader an opportunity
toabsorbthematerialinahands-onfashion,thebookincludesnumerousexamples
andexercisesofvariousdegreesofdifficulty.
Pedagogicalconsiderationscausedustorefrainfromstatingsomeoftheresults
in their full generality. In such cases, references to the appropriate literature are
provided for the interested reader. The notion of surface area is discussed in more
detailthaniscommoninanalysistexts.Usingadescriptivedefinition,weproveits
uniquenessonBorelsubsetsofsmoothandLipschitzmanifolds.
Itisdesirablethatthereaderbefamiliarwiththenotionofanintegralofacon-
tinuousfunctionofonevariableonanintervalpriortobeingexposedtothebasics
ofmeasuretheory.However,wedonotfeelthatthisprerequisitenecessarilyneeds
tobefulfilledinthecontextoftheRiemannintegral,whichweviewtobeprimarily
ofhistoricalinterest.ApossiblealternativeapproachisoutlinedinAppendix13.1.
This book is based on a series of lectures delivered by the authors at the De-
partment of Mathematics and Mechanics of St. Petersburg State University. The
majorityofthematerialinChaps.1–8approximatelycorrespondstothefourthand
fifthsemesteranalysisprogramformathematicsmajorsinourdepartment.Thema-
terialfromChaps.9–12andsomeotherpartsofthebookwaspreviouslyincluded
bytheauthorsinadvancedcoursesandlecturesinfunctionalanalysis.Someaddi-
tionalinformationispresentedinAppendices13.2–13.6.Appendix13.7,dedicated
tosmoothmappings,isincludedforthesakeofcompleteness.
Thereaderisexpectedtohavethenecessarymathematicalbackground.Thestu-
dentsenteringthefourthsemesterattheDepartmentofMathematicsandMechanics
ofSt.PetersburgStateUniversityarefamiliarwithmultivariablecalculusandbasic
linear algebra. This prerequisite material is used throughout the book without any
additionalexplanations.InChap.8,familiaritywiththebasicsofsmoothmanifold
theory is assumed. In Appendices 13.2 and 13.3, the rudiments of the theory of
metricspacesaretakenforgranted.
The authors have previously encountered texts where a definition or notation,
once introduced, is never repeated and is used without any further comments or
references many pages later. We believe that such manner of presentation, possi-
blyappropriateinmonographsofanencyclopedicnature,putstoomuchstrainon
the reader’s memory and attention span. Taking into account the fact that this is a
textbook intended for relatively inexperienced readers, many of whom will be en-
counteringthesubjectmatterforthefirsttime,theauthorsfinditusefultoinclude
somerepetitionsandreminders.However,theyareunabletomeasurethedegreeto
whichtheyhavesucceededinthisdirection.