Analysis on Polish Spaces and an Introduction to Optimal Transportation

356 Pages·2018·3.841 MB·English

by  D. J. H. Garling

LondonMathematicalSocietyStudentTexts89
Analysis on Polish Spaces
and an Introduction to Optimal Transportation
D. J. H. GARLING
EmeritusReaderinMathematicalAnalysis,UniversityofCambridge,
andFellowofStJohn’sCollege,Cambridge
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Title:AnalysisonPolishspacesandanintroductiontooptimal
transportation/D.J.H.Garling(UniversityofCambridge).
Othertitles:LondonMathematicalSocietystudenttexts;89.
Description:Cambridge,UnitedKingdom;NewYork,NY:CambridgeUniversity
Press,2018.|Series:LondonMathematicalSocietystudenttexts;89|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2017028186|ISBN9781108421577(hardback;alk.paper)|
ISBN1108421571(hardback;alk.paper)|ISBN9781108431767(pbk.;alk.paper)|
ISBN1108431763(pbk.;alk.paper)
Subjects:LCSH:Polishspaces(Mathematics)|Mathematicalanalysis.|
Transportationproblems(Programming)|Topology.
Classification:LCCQA611.28.G362018|DDC514/.32–dc23
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Contents
Introduction 1
PARTONE TOPOLOGICALPROPERTIES 7
1 GeneralTopology 9
1.1 TopologicalSpaces 9
1.2 Compactness 15
2 MetricSpaces 18
2.1 MetricSpaces 18
2.2 TheTopologyofMetricSpaces 21
2.3 Completeness:Tietze’sExtensionTheorem 24
2.4 MoreonCompleteness 27
2.5 TheCompletionofaMetricSpace 29
2.6 TopologicallyCompleteSpaces 31
2.7 Baire’sCategoryTheorem 33
2.8 LipschitzFunctions 35
3 PolishSpacesandCompactness 38
3.1 PolishSpaces 38
3.2 TotallyBoundedMetricSpaces 39
3.3 CompactMetrizableSpaces 41
3.4 LocallyCompactPolishSpaces 47
4 Semi-continuousFunctions 50
4.1 TheEffectiveDomainandProperFunctions 50
4.2 Semi-continuity 50
4.3 TheBre´zis–BrowderLemma 53
4.4 Ekeland’sVariationalPrinciple 54
v
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vi Contents
5 UniformSpacesandTopologicalGroups 56
5.1 UniformSpaces 56
5.2 TheUniformityofaCompactHausdorffSpace 59
5.3 TopologicalGroups 61
5.4 TheUniformitiesofaTopologicalGroup 64
5.5 GroupActions 66
5.6 MetrizableTopologicalGroups 67
6 Ca`dla`gFunctions 71
6.1 Ca`dla`gFunctions 71
6.2 TheSpace(D[0,1],d∞) 72
6.3 TheSkorohodTopology 73
6.4 TheMetricd 75
B
7 BanachSpaces 79
7.1 NormedSpacesandBanachSpaces 79
7.2 TheSpaceBL(X)ofBoundedLipschitzFunctions 82
7.3 IntroductiontoConvexity 83
7.4 ConvexSetsinaNormedSpace 86
7.5 LinearOperators 88
7.6 FiveFundamentalTheorems 91
7.7 ThePetalTheoremandDanesˇ’sDropTheorem 95
8 HilbertSpaces 97
8.1 Inner-productSpaces 97
8.2 HilbertSpace;NearestPoints 101
8.3 OrthonormalSequences;Gram–SchmidtOrthonormalization 104
8.4 OrthonormalBases 107
8.5 TheFre´chet–RieszRepresentationTheorem;Adjoints 108
9 TheHahn–BanachTheorem 112
9.1 TheHahn–BanachExtensionTheorem 112
9.2 TheSeparationTheorem 116
9.3 WeakTopologies 118
9.4 Polarity 119
9.5 WeakandWeak*TopologiesforNormedSpaces 120
9.6 Banach’sTheoremandtheBanach–AlaogluTheorem 124
9.7 TheComplexHahn–BanachTheorem 125
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Contents vii
10 ConvexFunctions 128
10.1 ConvexEnvelopes 128
10.2 ContinuousConvexFunctions 130
11 SubdifferentialsandtheLegendreTransform 133
11.1 DifferentialsandSubdifferentials 133
11.2 TheLegendreTransform 134
11.3 SomeExamplesofLegendreTransforms 137
11.4 TheEpisum 139
11.5 TheSubdifferentialofaVeryRegularConvexFunction 140
11.6 Smoothness 143
11.7 TheFenchel–RockafellerDualityTheorem 148
11.8 TheBishop–PhelpsTheorem 149
11.9 MonotoneandCyclicallyMonotoneSets 151
12 CompactConvexPolishSpaces 155
12.1 CompactPolishSubsetsofaDualPair 155
12.2 ExtremePoints 157
12.3 Dentability 160
13 SomeFixedPointTheorems 162
13.1 TheContractionMappingTheorem 162
13.2 FixedPointTheoremsofCaristiandClarke 165
13.3 Simplices 167
13.4 Sperner’sLemma 168
13.5 Brouwer’sFixedPointTheorem 170
13.6 Schauder’sFixedPointTheorem 171
13.7 FixedPointTheoremsofMarkovandKakutani 173
13.8 TheRyll–NardzewskiFixedPointTheorem 175
PARTTWO MEASURESONPOLISHSPACES 177
14 AbstractMeasureTheory 179
14.1 MeasurableSetsandFunctions 179
14.2 MeasureSpaces 182
14.3 ConvergenceofMeasurableFunctions 184
14.4 Integration 187
14.5 IntegrableFunctions 188
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viii Contents
15 FurtherMeasureTheory 191
15.1 RieszSpaces 191
15.2 SignedMeasures 194
15.3 M(X),L1andL∞ 196
15.4 TheRadon–NikodymTheorem 199
15.5 OrliczSpacesandLpSpaces 203
16 BorelMeasures 210
16.1 BorelMeasures,RegularityandTightness 210
16.2 RadonMeasures 214
16.3 BorelMeasuresonPolishSpaces 215
16.4 Lusin’sTheorem 216
16.5 MeasuresontheBernoulliSequenceSpace(cid:2)(N) 218
16.6 TheRieszRepresentationTheorem 222
16.7 TheLocallyCompactRieszRepresentationTheorem 225
16.8 TheStone–WeierstrassTheorem 226
16.9 ProductMeasures 228
16.10DisintegrationofMeasures 231
16.11TheGluingLemma 234
16.12HaarMeasureonCompactMetrizableGroups 236
16.13HaarMeasureonLocallyCompactPolishTopologicalGroups 238
17 MeasuresonEuclideanSpace 243
17.1 BorelMeasuresonRandRd 243
17.2 FunctionsofBoundedVariation 245
17.3 SphericalDerivatives 247
17.4 TheLebesgueDifferentiationTheorem 249
17.5 DifferentiatingSingularMeasures 250
17.6 DifferentiatingFunctionsinbv 251
0
17.7 Rademacher’sTheorem 254
18 ConvergenceofMeasures 257
18.1 TheNorm(cid:3).(cid:3) 257
TV
18.2 TheWeakTopologyw 258
18.3 ThePortmanteauTheorem 260
18.4 UniformTightness 264
18.5 Theβ Metric 266
18.6 TheProkhorovMetric 269
18.7 TheFourierTransformandtheCentralLimitTheorem 271
18.8 UniformIntegrability 276
18.9 UniformIntegrabilityinOrliczSpaces 278
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Contents ix
19 IntroductiontoChoquetTheory 280
19.1 Barycentres 280
19.2 TheLowerConvexEnvelopeRevisited 282
19.3 Choquet’sTheorem 284
19.4 Boundaries 285
19.5 PeakPoints 289
19.6 TheChoquetOrdering 291
19.7 Dilations 293
PARTTHREE INTRODUCTIONTOOPTIMAL
TRANSPORTATION 297
20 OptimalTransportation 299
20.1 TheMongeProblem 299
20.2 TheKantorovichProblem 300
20.3 TheKantorovich–RubinsteinTheorem 303
20.4 c-concavity 305
20.5 c-cyclicalMonotonicity 308
20.6 OptimalTransportPlansRevisited 310
20.7 Approximation 313
21 WassersteinMetrics 315
21.1 TheWassersteinMetricsW 315
p
21.2 TheWassersteinMetricW 317
1
21.3 W Compactness 318
1
21.4 W Compactness 320
p
21.5 W -Completeness 322
p
21.6 TheMallowsDistances 323
22 SomeExamples 325
22.1 StrictlySubadditiveMetricCostFunctions 325
22.2 TheRealLine 326
22.3 TheQuadraticCostFunction 327
22.4 TheMongeProblemonRd 329
22.5 StrictlyConvexTranslationInvariantCostsonRd 331
22.6 SomeStrictlyConcaveTranslation–InvariantCostsonRd 336
FurtherReading 339
Index 342
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Introduction
Analysisisconcernedwithcontinuityandconvergence.Investigationofthese
ideas led to the notions of topology and topological spaces. Once these
had been introduced, they became subjects in their own right, which were
investigated in fine detail to see how far the theory might lead (an excellent
illustrationofthisisgivenbythefascinatingbookbySteenandSeebach[SS]).
In practice, however, a great deal of analysis is concerned with what
happens on a very restricted class of topological spaces, namely, the Polish
spaces. A Polish space is a separable topological space whose topology is
defined by a complete metric. Important examples include Euclidean space,
pathwise-connectedRiemannianmanifolds,compactmetricspacesandsepa-
rableBanachspaces.
ThepurposeofthisbookistodevelopthestudyofanalysisonPolishspaces.
It consists of three parts. The first considers topological properties of Polish
spaces, and the second deals with the theory of measures on Polish spaces.
Inthethirdpart,wegiveanintroductiontothetheoryofoptimaltransportation.
Thismakesessentialuseoftheresultsofthefirsttwoparts,ormodificationsof
them.Itwas,infact,studyofoptimaltransportationthatledtotherealization
of how much its study required properties of Polish spaces, and measures
onthem.
There are three important advantages of restricting attention to Polish
spaces. First, many of the curious complications of the general topological
theory disappear. For example, a subspace of a separable topological space
neednotbeseparable,whereasasubspaceofaseparablemetricspaceisalways
separable.Secondly,theproofsofstandardresultsarefrequentlymucheasier
inthisrestrictedsetting.Forexample,Urysohn’slemmafornormaltopological
spaces is quite delicate, whereas it is very easy for metric spaces. Thirdly,
Polish spaces enjoy some very important properties. Thus it follows from
Alexandroff’s theorem that a topological space is a Polish space if and only
1
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2 Introduction
if it is homeomorphic to a G subset of the Hilbert cube H = [0,1]N, which
δ
is a compact metrizable space. From this, or directly, it follows that a Borel
measureonaPolishspaceistight(Ulam’stheorem:themeasureofaBorelset
canbeapproximatedfrombelowbythemeasuresofcompactsetscontainedin
it).ItalsomeansthatwecanpushforwardaBorelmeasureonaPolishspace
X to a Borel measure on a compact metric space containing X. This greatly
simplifies both the measure theory and also the construction of measures. In
fact,Ibelievethatalmostalltheprobabilitymeasuresthatariseinpracticeare
Borel measures on Polish spaces; one important exception, which we do not
considerorneed,isthetheoryofuniformcentrallimittheorems.
One major advantage of restricting attention to Polish spaces is that it
is not necessary to appeal to the axiom of choice. Instead, we proceed by
induction,usingtheaxiomofdependentchoice;wemakeaninfinitesequence
ofdecisions,eachpossiblydependentonwhathasgonebefore.
Inanalysis,thereareafewfundamentalresultswhichrequiretheaxiomof
choice.ThefirstisTychonoff’stheorem,whichstatesthatanarbitraryproduct
ofcompacttopologicalspaces,withtheproducttopology,iscompact.Wedo
notprovethis,oruseit.Ontheotherhand,wedoprove,anduse,thefactthat
acountableproductofcompactmetrizablespacesiscompactandmetrizable.
Secondly, there are two fundamental results of linear analysis which need
theaxiomofchoice,usingZorn’slemma.ThefirstoftheseistheHahn–Banach
theorem (together with the separation theorem). Using induction, we prove
weak forms of these, for separable normed spaces; this is sufficient for our
purposes.
But for completeness’ sake we also give the classical results, using Zorn’s
lemma;Herewefirstprovetheseparationtheorem,showingthatitessentially
dependsupontheconnectednessoftheunitcircleT,andthenderivetheHahn–
Banachtheoremfromit.
The other fundamental result which requires the axiom of choice is the
Krein–Mil’man theorem, which states that every weakly compact convex
subset K has an extreme point. Again, we only need, and use, the result in
thecasewhereKismetrizable,andweprovethiswithouttheaxiomofchoice.
The fact that we avoid using the axiom of choice suggests that the proofs
should, in some sense, be less abstract and more constructive. Unfortunately,
this is not the case; the arguments that are used are frequently indirect
(consider the collection of all sets with a particular property), so that for
example a typical Borel subset of a Polish space does not have a simple
description.
Let us now describe the contents of the three parts of this book in more
detail.
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