Real Analysis: Measure and Integration

358 Pages·2019·4.494 MB·English

by  Marat V. Markin

MaratV.Markin
RealAnalysis
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Marat V. Markin
Real Analysis
|
Measure and Integration
MathematicsSubjectClassification2010
28-01,28A10,28A12,28A15,28A20,28A25
Author
Prof.Dr.MaratV.Markin
CaliforniaStateUniversity,Fresno
DepartmentofMathematics
5245NorthBackerAvenue
Fresno,CA93740
USA
[email protected]
ISBN978-3-11-060097-1
e-ISBN(PDF)978-3-11-060099-5
e-ISBN(EPUB)978-3-11-059882-7
LibraryofCongressControlNumber:2019931612
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Withutmostappreciationtoallmyteachers.

Preface
Theauthordiscussesvaluelessmeasuresinpointlessspaces.
PaulHalmos
ThePurposeoftheBookandTargetedAudience
Thebookisintendedasatextforaone-semester Master’slevelgraduatecoursein
realanalysiswithemphasisonthemeasureandintegrationtheorytobetaughtwithin
theexistingconstraintsofthestandardfortheUnitedStatesgraduatecurriculum(fif-
teenweekswithtwoseventy-five-minutelecturesperweek).Realanalysis,being,as
arule,acorecourseineverygraduateprograminmathematics,isalsoofsignificant
interesttoawideraudienceofSTEM(science,technology,engineering,andmathe-
matics) graduate students or advanced undergraduates with a solid background in
proof-basedintermediateanalysis.
Book’sPhilosophy,Scope,andSpecifics
Thephilosophyofthebook,whichmakesitquitedistinctfrommanyexistingtextson
thesubject,isbasedontreatingtheconceptsofmeasureandintegrationstartingwith
the most general abstract setting and then introducing and studying the Lebesgue
measureandintegrationonthereallineasanimportantparticularcase.
Thebookconsistsofninechaptersandanappendixtakingthereaderfromtheba-
sicsetclasses,throughmeasures,outermeasuresandthegeneralprocedureofmea-
sureextensiondescribedinthecelebratedCarathéodory’sExtensionTheoremandap-
pliedtotheconstructoftheLebesgue–Stieltjesmeasuresasacentralparticularcase.
Itfurthertreatsmeasurablefunctionsandvarioustypesofconvergenceofsequencesof
suchbasedontheideaofmeasure,thetreatmentincludingtheclassicalLuzin’sand
Egorov’sTheoremaswellastheLebesgueandRieszTheorems.Thenthefundamen-
talsoftheabstractLebesgueintegrationandthebasiclimittheorems,suchasFatou’s
LemmaandLebesgue’sDominatedConvergenceTheorem,arefurnished,thediscourse
culminatingintothecomparisonoftheLebesgueandRiemannintegralsandcharac-
terizationoftheRiemannintegrablefunctions.
Chapter1outlinescertainnecessarypreliminaries,includingthefundamentalsof
metricspaces.ThecourseisdesignedtobetaughtstartingwithChapter2,Chapter1
beingreferredtowhenevertheneedarises,forinstancewhendealingwiththecon-
ceptsofupperandlowerlimitsofnumericorsetsequences,usingthepropertiesof
theinverseimageoperation,ordefiningtheBorelsets.
Chapter7isdedicatedtostudyingconvergenceinp-norm,L spaces,andtheba-
p
sicsofnormedvectorspaces.ItfurtherdemonstratesthedeficienciesoftheRiemann
https://doi.org/10.1515/9783110600995-201
VIII | Preface
integralrelativetoitsLebesguecounterpartandpreparesthestudentsformoread-
vancedcoursesinfunctionalanalysisandoperatortheory.
Chapter8isdedicatedtousingthenovelapproachbasedontheLebesguemea-
sure and integration theory machinery to develop a better understanding of differ-
entiation and extend the classical total change formula linking differentiation with
integrationtoasubstantiallywiderclassoffunctions.
Chapter9onsignedmeasurescanbeconsideredasa“bonus”chaptertobetaught
shouldthetimeconstraintsofaone-semestercoursepermit.
TheAppendixgivesaconcisetreatiseoftheAxiomofChoice,itsequivalents(the
HausdorffMaximalPrinciple,Zorn’sLemma,andZermello’sWell-OrderingPrinciple),
andorderedsets,whichisfundamentalforprovingthefamedVitaliTheoremonthe
existenceofanon-Lebesguemeasurablesetinℝ.
Beingdesignedasatexttobeusedinaclassroom,thebookconstantlycallsforthe
student’sactivelymasteringtheknowledgeofthesubjectmatter.Thereareproblems
attheendofeachchapter,startingwithChapter2andtotalingat125.Theseproblems
areindispensableforunderstandingthematerialandmovingforward.Manyimpor-
tantstatements,suchastheApproximationofBorelSetsProposition(Proposition4.6)
(Section4.8,Problem13),aregivenasproblemsandfrequentlyreferredtointhemain
body.Therearealso358exercisesthroughoutthetext,includingChapter1andthe
Appendix,whichrequireofthestudenttoproveorverifyastatementoranexample,
fillincertaindetailsinaproof,orprovideanintermediatesteporacounterexample.
Theyarealsoaninherentpartofthematerial.Moredifficultproblems,suchasSec-
tion8.6,Problem11,aremarkedwithanasterisk,manyproblemsandexercisesare
suppliedwith“existential”hints.
Thebookisgenerousonexamplesandcontainsnumerousremarksaccompany-
ingdefinitions,examples,andstatementstodiscusscertainsubtleties,raisequestions
onwhethertheconverseassertionsaretrue,wheneverappropriate,orwhetherthe
conditionsareessential.
Asamplydemonstratedbyexperience,studentstendtobetterrememberstate-
mentsbytheirnamesratherthanbynumbers.Thus,adistinctivefeatureofthebook
isthateverytheorem,proposition,corollary,andlemma,unlessalreadypossessinga
name,isendowedwithadescriptiveone,makingiteasiertoremember,which,inthis
author’shumbleopinion,isquiteabargainwhenthepriceforbetterunderstanding
andretentionofthematerialisalittleclumsinesswhilemakingalongerreference.
Eachstatementisreferredtobyitsnameandnotjustthenumber,e.g.,theCharacter-
izationofRiemannIntegrability(Theorem6.10),asopposedtomerelyTheorem6.10.
Withnopretenseonfurnishingthehistoryofthesubject,thetextprovidescertain
datesandlistseveryrelatednameasafootnote.