Introduction to Measure Theory and Integration

193 Pages·2011·0.961 MB·English

by  Ambrosio L., Da Prato G., Mennucci A.

10
APPUNTI
LECTURENOTES
LuigiAmbrosio,GiuseppeDaPratoandAndreaMennucci
ScuolaNormaleSuperiore
PiazzadeiCavalieri,7
56126Pisa,Italy
IntroductiontoMeasureTheoryandIntegration
Luigi Ambrosio, Giuseppe Da Prato
and Andrea Mennucci
Introduction
to Measure Theory
and Integration
(cid:2)c 2011ScuolaNormaleSuperiorePisa
ISBN:978-88-7642-385-7
e-ISBN:978-88-7642-386-4
Contents
Preface ix
Introduction xi
1 Measurespaces 1
1.1 Notationandpreliminaries . . . . . . . . . . . . . . . . 1
1.2 Rings,algebrasandσ–algebras . . . . . . . . . . . . . . 2
1.3 Additiveandσ–additivefunctions . . . . . . . . . . . . 4
1.4 Measurablespacesandmeasurespaces . . . . . . . . . . 7
1.5 Thebasicextensiontheorem . . . . . . . . . . . . . . . 8
1.5.1 Dynkinsystems . . . . . . . . . . . . . . . . . . 9
1.5.2 Theoutermeasure . . . . . . . . . . . . . . . . 11
1.6 TheLebesguemeasureinR . . . . . . . . . . . . . . . 14
1.7 Innerandouterregularityofmeasuresonmetricspaces . 18
2 Integration 23
2.1 Inverseimageofafunction . . . . . . . . . . . . . . . . 23
2.2 MeasurableandBorelfunctions . . . . . . . . . . . . . 24
2.3 Partitionsandsimplefunctions . . . . . . . . . . . . . . 25
2.4 IntegralofanonnegativeE–measurablefunction . . . . 27
2.4.1 Integralofsimplefunctions . . . . . . . . . . . 27
2.4.2 Therepartitionfunction . . . . . . . . . . . . . 28
2.4.3 Thearchimedeanintegral . . . . . . . . . . . . . 31
2.4.4 Integralofanonnegativemeasurablefunction . . 32
2.5 Integraloffunctionswithavariablesign . . . . . . . . . 35
2.6 Convergenceofintegrals . . . . . . . . . . . . . . . . . 36
2.6.1 UniformintegrabilityandVitaliconvergence
theorem . . . . . . . . . . . . . . . . . . . . . . 38
2.7 AcharacterizationofRiemannintegrablefunctions . . . 39
vi LuigiAmbrosio,GiuseppeDaPratoandAndreaMennucci
3 Spacesofintegrablefunctions 45
3.1 SpacesLp(X,E,μ)and Lp(X,E,μ) . . . . . . . . . . 45
3.2 The Lp norm . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.1 Ho¨lderandMinkowskiinequalities . . . . . . . 48
3.3 Convergencein Lp(X,E,μ)andcompleteness . . . . . 49
3.4 Thespace L∞(X,E,μ) . . . . . . . . . . . . . . . . . . 52
3.5 Densesubsetsof Lp(X,E,μ) . . . . . . . . . . . . . . 56
4 Hilbertspaces 61
4.1 Scalarproducts,pre-HilbertandHilbertspaces . . . . . 61
4.2 Theprojectiontheorem . . . . . . . . . . . . . . . . . . 63
4.3 Linearcontinuousfunctionals. . . . . . . . . . . . . . . 66
4.4 Besselinequality,Parsevalidentityandorthonormal
systems . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5 HilbertspacesonC . . . . . . . . . . . . . . . . . . . . 70
5 Fourierseries 73
5.1 PointwiseconvergenceoftheFourierseries . . . . . . . 75
5.2 Completenessofthetrigonometricsystem . . . . . . . . 79
5.3 UniformconvergenceoftheFourierseries . . . . . . . . 80
6 Operationsonmeasures 83
6.1 TheproductmeasureandFubini–Tonellitheorem . . . . 83
6.2 TheLebesguemeasureonRn . . . . . . . . . . . . . . . 87
6.3 Countableproducts . . . . . . . . . . . . . . . . . . . . 90
6.4 Comparisonofmeasures . . . . . . . . . . . . . . . . . 94
6.5 Signedmeasures . . . . . . . . . . . . . . . . . . . . . 101
6.6 MeasuresinR . . . . . . . . . . . . . . . . . . . . . . . 105
6.7 ConvergenceofmeasuresonR . . . . . . . . . . . . . . 107
6.8 Fouriertransform . . . . . . . . . . . . . . . . . . . . . 112
6.8.1 Fouriertransformofameasure . . . . . . . . . . 113
7 Thefundamentaltheoremoftheintegralcalculus 119
8 Measurabletransformations 129
8.1 Imagemeasure . . . . . . . . . . . . . . . . . . . . . . 129
8.2 Changeofvariablesinmultipleintegrals . . . . . . . . . 130
8.3 ImagemeasureofLn byaC1 diffeomorphism . . . . . 131
A 137
A.1 Continuity and differentiability of functions depending
onaparameter. . . . . . . . . . . . . . . . . . . . . . . 137
vii IntroductiontoMeasureTheoryandIntegration
A.2 Thedualspaceofcontinuousfunctions. . . . . . . . . . 139
References 183
Preface
This textbook collects the notes for an introductory course in measure
theoryandintegrationtaughtbytheauthorstoundergraduatestudentsof
ScuolaNormaleSuperioreinthelast10years.
The goal of the course was to present, in a quick but rigorous way,
themodernpointofviewonmeasuretheoryandintegration,puttingLe-
besgues theory in Rn into a more general context and presenting the ba-
sicapplicationstoFourierseries,calculusandrealanalysis. Thetextcan
also pave the way to more advanced courses in probability, stochastic
processesorgeometricmeasuretheory.
Prerequisitesforthebookareabasicknowledgeofcalculusinoneand
severalvariables,metricspacesandlinearalgebra.
All results presented here, as well as their proofs, are classical. We
claim some originality only in the presentation and in the choice of the
exercises. Detailedsolutionstotheexercisesareprovidedinthefinalpart
ofthebook.
Pisa,July2011
LuigiAmbrosio,GiuseppeDaPrato
andAndreaMennucci
Introduction
Thiscourseconsistsofanintroductiontothemoderntheoriesofmeasure
andofintegration. Historically,thishasbeenmotivatedbythenecessity
togobeyondtheclassicaltheoryofRiemann’sintegration,usuallytaught
in elementary Calculus courses on the real line. It is therefore useful to
describethereasonsthatmotivatethisextension.
(1)Itisnotpossibletogiveasimple,handy,characterizationoftheclass
ofRiemann’sintegrablefunction,withinRiemann’stheory. Thisisindeed
possiblewithinthestrongertheory, dueessentiallytoLebesgue, that we
aregoingtointroduce.
(2) The extensions of Riemann’s theory to multiple integrals are very
cumbersome. This extension, useful to compute areas, volumes, etc., is
knownasPeano–Jordantheory,anditissometimestaughtinelementary
coursesofintegrationinmorethanonevariable. Inadditiontothat, im-
portantheuristicprincipleslikeCavalieri’sonecanbeprovedonlyunder
technical and basically unnecessary regularity assumptions on the do-
mainsofintegration.
(3) Many constructive processes typical of Analysis (limits, series, in-
tegrals depending on a parameter, etc.) cannot be handled well within
Riemann’stheoryofintegration. Forinstance,thefollowingstatementis
true (it is a particular case of the so-called dominated convergence the-
orem):
Theorem1. Let f : [−1,1] → R be continuous functions pointwise
h
converging to a continuous function f. Assume the existence of a con-
stant M satisfying|f (x)| ≤ M forall x ∈ [−1,1]andallh ∈ N. Then
h
(cid:2) (cid:2)
1 1
lim f (x)dx = f(x)dx.
h
h→∞ −1 −1
EventhoughthisstatementmakesperfectlysensewithinRiemann’sthe-
ory, any attempt to prove this result within the theory (try, if you don’t