Gaussian Measures in Hilbert Space
Construction and Properties
Alexander Kukush
First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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AbbreviationsandNotation
Chapter1.GaussianMeasuresinEuclideanSpace ...........1
1.1.Thechangeofvariablesformula......................1
1.2.InvarianceofLebesguemeasure......................4
1.3.Absenceofinvariantmeasureininfinite-dimensionalHilbertspace..9
1.4.Randomvectorsandtheirdistributions..................10
1.4.1.Randomvariables............................11
1.4.2.Randomvectors.............................12
1.4.3.Distributionsofrandomvectors....................14
1.5.GaussianvectorsandGaussianmeasures.................17
1.5.1.CharacteristicfunctionsofGaussianvectors.............17
1.5.2.ExpansionofGaussianvector.....................20
1.5.3.SupportofGaussianvector.......................22
1.5.4.GaussianmeasuresinEuclideanspace................23
Chapter2.GaussianMeasurein
2.1.Space R
2.1.1.Metricon
2.1.2.Borelandcylindricalsigma-algebrascoincide............30
2.1.3.Weighted l2 space............................31
2.2.Productmeasurein R∞
2.2.1.Kolmogorovextensiontheorem....................34
2.2.2.Constructionofproductmeasureon B (R∞ ) .............36
2.2.3.Propertiesofproductmeasure.....................38
2.3.StandardGaussianmeasurein R∞ .....................42
2.3.1.Alternativeproofofthesecondpartoftheorem2.4.........45
2.4.ConstructionofGaussianmeasurein l2 ..................46
Chapter3.BorelMeasuresinHilbertSpace ................51
3.1.Classesofoperatorsin H ..........................51
3.1.1.Hilbert–Schmidtoperators.......................52
3.1.2.Polardecomposition...........................55
3.1.3.Nuclearoperators............................57
3.1.4. S-operators................................62
3.2.PettisandBochnerintegrals........................68
3.2.1.Weakintegral..............................68
3.2.2.Strongintegral..............................69
3.3.BorelmeasuresinHilbertspace......................75
3.3.1.Weakandstrongmoments.......................75
3.3.2.ExamplesofBorelmeasures......................78
3.3.3.Boundednessofmomentform.....................83
Chapter4.ConstructionofMeasurebyitsCharacteristic Functional ......................................89
4.1.Cylindricalsigma-algebrainnormedspace................89 4.2.Convolutionofmeasures..........................93
4.3.Propertiesofcharacteristicfunctionalsin H ...............96
4.4. S -topologyin H ..............................99
4.5.Minlos–Sazonovtheorem..........................102
Chapter5.GaussianMeasureofGeneralForm
5.1.CharacteristicfunctionalofGaussianmeasure..............111
5.2.DecompositionofGaussianmeasureandGaussianrandomelement..114
5.3.SupportofGaussianmeasureanditsinvariance.............117
5.4.WeakconvergenceofGaussianmeasures.................125
5.5.ExponentialmomentsofGaussianmeasureinnormedspace......129
5.5.1.Gaussianmeasuresinnormedspace..................129
5.5.2.Fernique’stheorem...........................133
Chapter6.EquivalenceandSingularityofGaussianMeasures ...143
6.1.Uniformlyintegrablesequences......................143
6.2.Kakutani’sdichotomyforproductmeasureson R∞ ...........145
6.2.1.Generalpropertiesofabsolutelycontinuousmeasures........145
6.2.2.Kakutani’stheoremforproductmeasures...............148
6.2.3.DichotomyforGaussianproductmeasures..............152
6.3.Feldman–HájekdichotomyforGaussianmeasureson H ........155
6.3.1.ThecasewhereGaussianmeasureshaveequalcorrelation operators.....................................155
6.3.2.NecessaryconditionsforequivalenceofGaussianmeasures....158
6.3.3.CriterionforequivalenceofGaussianmeasures...........165
6.4.Applicationsinstatistics..........................169
6.4.1.EstimationandhypothesistestingformeanofGaussianrandom element.....................................169
6.4.2.Estimationandhypothesistestingforcorrelationoperatorof centeredGaussianrandomelement......................173
Chapter7.Solutions
...............................179
7.1.SolutionsforChapter1...........................179
7.2.SolutionsforChapter2...........................193
7.2.1.GeneralizedKolmogorovextensiontheorem.............196
7.3.SolutionsforChapter3...........................202
7.4.SolutionsforChapter4...........................211
7.5.SolutionsforChapter5...........................217
7.6.SolutionsforChapter6...........................227
SummarizingRemarks ..............................235
Foreword
Thestudyofmoderntheoryofstochasticprocesses,infinite-dimensionalanalysis andMalliavincalculusisimpossiblewithoutasolidknowledgeofGaussian measuresoninfinite-dimensionalspaces.Inspiteoftheimportanceofthistopicand theabundanceofliteratureavailableforexperiencedresearchers,thereisnotextbook suitableforstudentsforafirstreading.
Thepresentmanualisanexcellentget-to-knowcourseinGaussianmeasureson infinite-dimensionalspaces,whichhasbeengivenbytheauthorformanyyearsat theFacultyofMechanics&MathematicsofTarasShevchenkoNationalUniversityof Kyiv,Ukraine.Thepresentationofthematerialiswellthoughtout,andthecourseis self-contained.Afterreadingthebookitmayseemthatthetopicisverysimple.But thatisnottrue!Apparentsimplicityisachievedbycarefulorganizationofthebook. ForexpertsandPhDstudentshavingexperienceininfinite-dimensionalanalysis,I prefertorecommendthemonographV.I.Bogachev, GaussianMeasures (1998).But forfirstacquaintancewiththetopic,Irecommendthisnewmanual.
Prerequisitesforthebookareonlyabasicknowledgeofprobabilitytheory,linear algebra,measuretheoryandfunctionalanalysis.Theexpositionissupplementedwith abulkofexamplesandexerciseswithsolutions,whichareveryusefulforunassisted workandcontrolofstudiedmaterial.
Inthisbook,manydelicateandimportanttopicsofinfinite-dimensionalanalysis areanalyzedindetail,e.g.Borelandcylindricalsigma-algebrasin infinite-dimensionalspaces,BochnerandPettisintegrals,nuclearoperatorsandthe topologyofnuclearconvergence,etc.Wepresentthecontentsofthebook, emphasizingplaceswherefinite-dimensionalresultsneedreconsideration (everywhereexceptChapter1).
–Chapter1. Gaussiandistributionsonafinite-dimensionalspace. Thechapteris preparatorybutnecessary.Lateron,manyanalogieswithfinite-dimensionalspacewill begiven,andtheplaceswillbevisiblewhereanewtechniqueisneeded.
–Chapter2. Space R∞ ,Kolmogorovtheoremabouttheexistenceofprobability measure,productmeasures,Gaussianproductmeasures,Gaussianproductmeasures in l2 space. Afterreadingthechapter,thestudentwillstarttounderstandthaton infinite-dimensionalspacethereareseveralwaystodefineasigma-algebra(luckily, inourcaseBorelandcylindricalsigma-algebrascoincide).Moreover,itwillbecome clearthatinfinite-dimensionalLebesguemeasuredoesnotexist,henceconstruction ofmeasurebymeansofdensityneedsreconsideration.
–Chapter3. BochnerandPettisintegrals,Hilbert–Schmidtoperatorsandnuclear operators,strongandweakmoments. Thechapterisapreparationforthedefinition oftheexpectationandcorrelationoperatorofGaussian(orevenarbitrary)random element.Weseethatitisnotsoeasytointroduceexpectationofarandomelement distributedinHilbertorBanachspace.Asopposedtofinite-dimensionalspace,itis notenoughjusttointegrateoverbasisvectorsandthenaugmenttheresultsinasingle vector.
–Chapter4. Characteristicfunctionals,Minlos–Sazonovtheorem. Oneofthemost importantmethodstoinvestigateprobabilitymeasuresonfinite-dimensionalspaceis themethodofcharacteristicfunctions.Aswell-knownfromthecourseofprobability theory,thesewillbeallcontinuouspositivedefinitefunctionsequaltooneatzero,and onlythem.Oninfinite-dimensionalspacethisisnottrue.Forthestatement“theyand onlythem”,continuityinthetopologyofnuclearconvergenceisrequired,andthis topologyisexplainedindetail.
–Chapter5. GeneralGaussianmeasures. Basedonresultsofpreviouschapters, weseethenecessaryandsufficientconditionsthathavetobesatisfiedbythe characteristicfunctionalofaGaussianmeasureinHilbertspace.Werealizethatwe haveusedalltheknowledgefromChapters2–4(concerningintegrationofrandom elements,aboutHilbert–Schmidtandnuclearoperators,Minlos–Sazonovtheorem, etc.).Wenoticethatfortheeigenbasisofthecorrelationoperator,aGaussian measureisjustaproductmeasurewhichweconstructedinChapter2.Thisseems natural;butonourwayitwasimpossibletodiscardanysinglestepwithoutlossof mathematicalrigor.Inthischapter,Fernique’stheoremaboutfinitenessofan exponentialmomentofthenormofaGaussianrandomelementisprovedandthe criterionfortheweakconvergenceofGaussianmeasuresisstated.
–Chapter6. Equivalenceandmutualsingularityofmeasures. Here,Kakutani’s theoremisprovenabouttheequivalenceoftheinfiniteproductofmeasures.Aswe sawinthepreviouschapter,GaussianmeasuresonHilbertspacesareproduct measures,inaway.Therefore,asaconsequenceofgeneraltheory,wegetacriterion fortheequivalenceofGaussianmeasures(Feldman–Hájektheorem).Theobtained resultsareappliedtoproblemsofinfinite-dimensionalstatistics.Oneshouldbe
carefulhere,asduetotheabsenceoftheinfinite-dimensionalLebesguemeasure,the Radon–Nikodymdensityshouldbewrittenw.r.t.oneoftheGaussianmeasures.
Theauthorofthisbook,ProfessorA.G.Kukush,hasbeenworkingattheFaculty ofMechanics&MathematicsofTarasShevchenkoNationalUniversityfor40years. Heisanexcellentteacherandafamousexpertinstatisticsandprobabilitytheory.In particular,heusedtogivelecturestostudentsofmathematicsandstatisticson MeasureTheory,FunctionalAnalysis,StatisticsandEconometrics.Asastudent,I wasluckytoattendhisfascinatingcourseoninfinite-dimensionalanalysis.
AndreyP
ILIPENKO LeadingResearcherattheInstituteofMathematics ofUkrainianNationalAcademyofSciences, ProfessorofMathematicsattheNationalTechnicalUniversity ofUkraine,“IgorSikorskyKyivPolytechnicInstitute” August2019
Preface
Thisbookiswrittenforgraduatestudentsofmathematicsandmathematical statisticswhoknowalgebra,measuretheoryandfunctionalanalysis(generalized functionsarenotusedhere);theknowledgeofmathematicalstatisticsisdesirable onlytounderstandsection6.4.Thetopicofthisbookcanbeconsideredas supplementarychaptersofmeasuretheoryandliesbetweenmeasuretheoryandthe theoryofstochasticprocesses;possibleapplicationsareinfunctionalanalysisand statisticsofstochasticprocesses.For20years,theauthorhasbeengivingaspecial course“GaussianMeasures”atTarasShevchenkoNationalUniversityofKyiv, Ukraine,andin2018–2019,preliminaryversionsofthisbookhavebeenusedasa textbookforthiscourse.
Thereareexcellenttextbooksandmonographsonrelatedtopics,suchas Gaussian MeasuresinBanachSpaces [KUO75], GaussianMeasures [BOG98]and Probability DistributionsonBanachSpaces [VAK87].WhydidIwritemyowntextbook?
Inthe1970s,IstudiedattheFacultyofMechanicsandMathematicsofTaras ShevchenkoNationalUniversityofKyiv,atthattimecalledKievStateUniversity. ThereIattendedunforgettablelecturesgivenbyProfessorsAnatoliyYa. Dorogovtsev(calculusandmeasuretheory),LevA.Kaluzhnin(algebra),MykhailoI. Yadrenko(probabilitytheory),MyroslavL.Gorbachuk(functionalanalysis)and YuriyM.Berezansky(spectraltheoryoflinearoperators).MyPhDthesiswas supervisedbyfamousstatisticianA.Ya.Dorogovtsevanddealtwiththeweak convergenceofmeasuresoninfinite-dimensionalspaces.Forlongtime,Iwasa memberoftheresearchseminar“Stochasticprocessesanddistributionsinfunctional spaces”headedbyclassicsofprobabilitytheoryAnatoliyV.SkorokhodandYuriyL. Daletskii.Myseconddoctoralthesiswasaboutasymptoticpropertiesofestimators forparametersofstochasticprocesses.Thus,Iamsomewhattiedupwithmeasures oninfinite-dimensionalspaces.
In1979,Kuo’sfascinatingtextbookwastranslatedintoRussian.Inspiredbythis book,IstartedtogivemylecturesonGaussianmeasuresforgraduatestudents.The subjectseemedhighlytechnicalandextremelydifficult.Idecidedtocreate somethinglikeacomicbookonthistopic,inparticulartodividelengthyproofsinto smallunderstandablestepsandexplaintheideasbehindcomputations.
Itisimpossibletostudymathematicalcourseswithoutsolvingproblems.Each sectionendswithseveralproblems,someofwhichareoriginalandsomearetaken fromdifferentsources.Aseparatechaptercontainsdetailedsolutionstoallthe problems.
Acknowledgments
IwouldliketothankmycolleaguesatTarasShevchenkoNationalUniversityof Kyivwhosupportedmyproject,especiallyYuliyaMishura,OleksiyNesterenkoand IvanFeshchenko.AlsoIwishtothankmystudentsofdifferentgenerationswho followedupontheideasofthematerialandhelpedmetoimprovethepresentation.I amgratefultoFedorNazarov(KentStateUniversity,USA)whocommunicatedthe proofoftheorem3.9.Inparticular,IamgratefultoOksanaChernovaandAndrey Frolkinforpreparingthemanuscriptforpublication.IthankSergiyShklyarforhis valuablecomments.
MywifeMariyadeservesthemostthanksforherencouragementandpatience.
AlexanderK UKUSH Kyiv,Ukraine September2019
ThetheoryofGaussianmeasuresliesonthejunctionoftheoryofstochastic processes,functionalanalysisandmathematicalphysics.Possibleapplicationsarein quantummechanics,statisticalphysics,financialmathematicsandotherbranchesof science.Inthisfield,theideasandmethodsofprobabilitytheory,nonlinearanalysis, geometryandtheoryoflinearoperatorsinteractinanelegantandintriguingway.
TheaimofthisbookistoexplaintheconstructionofGaussianmeasureinHilbert space,presentitsmainpropertiesandalsooutlinepossibleapplicationsinstatistics.
Chapter1dealswithEuclideanspace,wheretheinvarianceofLebesguemeasure isexplainedandGaussianvectorsandGaussianmeasuresareintroduced.Their propertiesarestatedinsuchaformthat(lateron)theycanbeextendedtothe infinite-dimensionalcase.Furthermore,itisshownthatonaninfinite-dimensional Hilbertspacethereisnonon-trivialmeasure,whichisinvariantunderalltranslations (thesameconcerninginvarianceunderallunitaryoperators);henceonsuchaspace thereisnomeasureanalogoustotheLebesgueone.
InChapter2,aproductmeasureisconstructedonthesequencespace R∞ based onKolmogorovextensiontheorem.ForstandardGaussianmeasure μ on R∞ , Kolmogorov–Khinchincriterionisestablished.Inparticular,itisshownthat μ is concentratedoncertainweightedsequencespaces l2,a ,andbasedonisometry between l2,a and l2 ,aGaussianproductmeasureisconstructedonthelattersequence space.
Chapter3introducesimportantclassesofoperatorsinaseparable infinite-dimensionalHilbertspace H ,inparticular S -operators,i.e.self-adjoint, positiveandnuclearones.Theorem3.9showsthattheconvergenceof S -operatorsis equivalenttocertainconvergenceofcorrespondingquadraticforms.Alsotheweak (Pettis)andstrong(Bochner)integralsaredefinedforafunctionvaluedinaBanach space.
Borelprobabilitymeasureson H andanormedspace X arestudiedwithexamples. Theboundednessofmomentformsofsuchmeasuresisshown,withsimpleproof basedontheclassicalBanach–Steinhaustheorem.Corollary3.3andremark3.8give mildconditionsfortheexistenceofmeanvalueofaprobabilitymeasure μ asPettis integral,andiftheunderlyingspaceisaseparableBanachspace B and μ hasastrong firstmoment,thenitsmeanvalueexistsasBochnerintegral.
InChapter4,propertiesofcharacteristicfunctionalsofBorelprobabilitymeasures on H arestudied.Aspeciallineartopology, S -topology,isintroducedin H witha neighborhoodsystemconsistingofellipsoids.ClassicalMinlos–Sazonovtheoremis provenandproperlyextendsBochner’stheoremfrom Rn to H .AccordingtoMinlos–Sazonovtheorem,thecharacteristicfunctionalofaBorelprobabilitymeasureson H shouldbecontinuousin S -topology.Apartofproofofthistheorem(seelemma4.9) suggeststhewaytoconstructaprobabilitymeasurebyitscharacteristicfunctional.
InChapter5,theorem5.1usestheMinlos–Sazonovtheoremtodescribea Gaussianmeasureon H ofgeneralform.Itturnsoutthatthecorrelationoperatorof suchameasureisalwaysan S -operator.ItisshownthateachGaussianmeasureon H isjustaproductofone-dimensionalGaussianmeasuresw.r.t.theeigenbasisofthe correlationoperator.Thus,everyGaussianmeasureon H canbeconstructedalong theway,asdemonstratedinChapter2.
ThesupportofGaussianmeasureisstudied.ItisshownthatacenteredGaussian measureisinvariantunderquitearichgroupoflineartransforms(seetheorem5.5). Hence,aGaussianmeasureinHilbertspacecanbeconsideredasanaturalinfinitedimensionalanalogueof(invariant)Lebesguemeasure.
AcriterionfortheweakconvergenceofGaussianmeasuresisstated,where(dueto theorem3.9)werecognizetheconvergenceofcorrelationoperatorsinnuclearnorm.
Insection5.5,westudyGaussianmeasuresonaseparablenormedspace X Importantexample5.3showsthataGaussianstochasticprocessgeneratesameasure onthepathspace Lp [0,T ],henceincase p =2,weobtainaGaussianmeasureon Hilbertspace.Lemma5.9presentsacharacterizationofGaussianrandomelementin X
ThefamoustheoremofFerniqueisproven,whichstatesthatcertainexponential momentsofaGaussianmeasureon X arefinite.Inparticular,everyGaussianmeasure onaseparableBanachspace B hasmeanvalueasBochnerintegralanditscorrelation operatoriswell-defined.Theorem5.10derivestheconvergenceofmomentsofweakly convergentGaussianmeasures.
InChapter6,Kakutani’sremarkabledichotomyforproductmeasureson R∞ is proven.Inparticular,twosuchproductmeasureswithabsolutelycontinuous
componentsareeitherabsolutelycontinuousormutuallysingular.Thisimpliesthe dichotomyforGaussianmeasureson R∞ :twosuchmeasuresareeitherequivalentor mutuallysingular.Section6.3provesthefamousFeldman–Hájekdichotomyfor Gaussianmeasureson H ,andincaseofequivalentmeasures,expressionsfor Radon–Nikodymderivativesareprovided.
Insection6.4,theresultsofChapter6areappliedinstatistics.Basedonasingle observationofGaussianrandomelementin H ,weconstructunbiasedestimatorsfor itsmeanandforparametersofitscorrelationoperator;alsowecheckahypothesis aboutthemeanandthecorrelationoperator(thelatterhypothesisisinthecasewhere theGaussianelementiscentered).Inviewofexample5.3with p =2,thesestatistical procedurescanbeusedforasingleobservationofaGaussianprocessonfinitetime interval.
Thebookisaimedforadvancedundergraduatestudentsandgraduatestudentsin mathematicsandstatistics,andalsofortheoreticallyinterestedstudentsfromother disciplines,sayphysics.
Prerequisites forthebookarecalculus,algebra,measuretheory,basicprobability theoryandfunctionalanalysis(wedonotusegeneralizedfunctions).Insection6.4, theknowledgeofbasicmathematicalstatisticsisrequired.
Somewordsabout thestructureofthebook :wepresenttheresultsinlemmas, theorems,corollariesandremarks.Allstatementsareproven.Importantand illustrativeexamplesaregiven.Furthermore,eachsectionendswithalistof problems.DetailedsolutionstotheproblemsareprovidedinChapter7.
Theabbreviationsandnotationusedinthebookaredefinedinthecorresponding chapters;anoverviewofthemisgiveninthefollowinglist.
AbbreviationsandNotation
a.e.almosteverywherew.r.t.Lebesguemeasure a.s.almostsurely cdfcumulativedistributionfunction pdfprobabilitydensityfunction i.i.d.independentandidenticallydistributed(randomvariablesor vectors)
r.v.randomvariable
LHSleft-handside
RHSright-handside
MLEmaximumlikelihoodestimator
|A| numberofpointsinset A
Ac complementofset A
A closureofset A
TB imageofset B undertransformation T
T 1 A preimageofset A undertransformation T
x , A transposedvectorandtransposedmatrix,respectively R extendedrealline,i.e. R = R ∪{−∞, +∞}
Rn×m spaceofreal n × m matrices
B (x,r ), B (x,r ) openandclosedball,respectively,centeredat x withradius r> 0 inametricspace
f+ positivepartoffunction f , f+ =max(f, 0) f negativepartoffunction f , f = min(f, 0)
δij Kroneckerdelta, δij =1 if i = j ,and δij =0, otherwise
an ∼ bn {an } isequivalentto {bn } as n →∞,i.e. an /bn → 1 as n →∞
C (X ) spaceofallrealcontinuousfunctionson X
R∞ spaceofallrealsequences
B (X ) Borelsigma-algebraonmetric(ortopological)space X
λm Lebesguemeasureon Rm
Sm sigma-algebraofLebesguemeasurablesetson Rm
IA indicatorfunction,i.e. IA (x)=1 if x ∈ A,else IA (x)=0 μT 1 measureinducedbymeasurabletransformation T basedon measure μ,i.e. (μT 1 )(A)= μ(T 1 A),foreachmeasurable set A
L(X,μ) spaceofLebesgueintegrablefunctionson X w.r.t.measure μ f = g (mod μ)functions f and g areequalalmosteverywherew.r.t.measure μ
δx Diracmeasureatpoint x, δx (B )= IB (x)
ν μ signedmeasure ν isabsolutelycontinuousw.r.t.measure μ dν dμ theRadon–Nikodymderivativeof ν w.r.t. μ
ν ∼ μ measures ν and μ areequivalent
ν ⊥ μ signedmeasure ν andmeasure μ aremutuallysingular (x,y ) innerproductofvectors x and y inEuclideanorHilbertspace x Euclideannormofvector x A Euclideannormofmatrix A, A =sup x=0 Ax x Im theidentitymatrixofsize m rk (S ) rankofmatrix S Pn projectiveoperator, Pn x =(x1 ,...,xn ) , x ∈ R∞ √A squarerootofpositivesemidefinitematrix A,itispositive semidefiniteaswellwith (√A)2 = A
x,x∗ or x∗ ,x valueoffunctional x∗ atvector x I theidentityoperator
L(X ) spaceoflinearboundedoperatorsonnormedspace X R(A) rangeofoperator A, R(A)= {y : ∃x,y = Ax}
L⊥ orthogonalcomplementtoset L
L2 [a,b] Hilbertspaceofsquareintegrablerealfunctionswithinner product (x,y )= b a x(t)y (t)dt,thelatterisLebesgueintegral lp spaceofrealsequences x =(xn )∞ 1 withnorm x p = ( ∞ n=1 |xn |p )1/p if 1 ≤ p< ∞,and x ∞ =supn≥1 |xn | if p = ∞.For p =2, l2 isHilbertspacewithinner product (x,y )= ∞ 1 xn yn
l2,a weighted l2 space span(M ) spanofset M, i.e.,setofallfinitelinearcombinationsofvectors from M
ˆ An cylinderin R∞ withbase An ∈B (Rn )
A = A L(X ) operatornormoflinearboundedoperator A, A =sup x=0 Ax x
A∗ adjointoperator
√B = B 1/2 squarerootofself-adjointpositiveoperator B
|A| modulusofcompactoperator A, |A| =(A∗ A)1/2
AbbreviationsandNotationxxi
A 1 nuclearnormofoperator A
A 2 Hilbert–Schmidtnormofoperator A
An ⇒ A operators An uniformlyconvergetooperator A
S0 (H ) classoffinite-dimensionaloperatorson H
S1 (H ) classofnuclearoperatorson H
S2 (H ) classofHilbert–Schmidtoperatorson H
S∞ (H ) classofcompactoperatorson H
LS (H ) classof S -operatorson H
A ≥ 0 operator A ispositive,i.e. (Ax,x) ≥ 0 forall x
A ≥ B comparisoninLoewnerorderofself-adjointoperators: A B is positiveoperator
n k =1 Ak Cartesianproductofsets A1 ,...,An
n
k =1 μk productofmeasures μ1 ,...,μn
∞
k =1 μk productmeasureon R∞ oronHilbertspace
mμ meanvalueofmeasure μ
Cov (μ) variance-covariancematrixofmeasure μ on Rn
ϕμ or ˆ μ characteristicfunction(orfunctional)ofmeasure μ
Aμ operatorofsecondmomentofmeasure μ
Sμ correlationoperatorofmeasure μ
σn (z1 ,...,zn ) weakmomentsoforder n ofBorelprobabilitymeasureon H
μX distributionofrandomvector X orrandomelement X , μX (B )= P(X ∈ B ) forallBorelsets B
ϕX characteristicfunction(functional)ofrandomvector (element) X
E X expectationofrandomvector(element) X
D X varianceofrandomvariable X
Cov (X ) variance-covariancematrixofrandomvector X
X d = Y randomvectors(elements) X and Y areidenticallydistributed
N (m,σ 2 ) Gaussiandistributionwithmean m andvariance σ 2 , σ ≥ 0
N (m,S ) Gaussiandistributionon Rn (oron H )withmeanvalue m and variance-covariancematrix(orcorrelationoperator) S d −→ convergenceindistributionofrandomelements
GaussianMeasuresinEuclideanSpace
1.1.Thechangeofvariablesformula
Let (X,S,μ) beameasurespace,i.e. X isanon-emptyset, S isasigma-algebraon X and μ isameasureon S .Consideralsoameasurablespace (Y,F ),i.e. Y isanother non-emptysetand F isasigma-algebraon Y. Let T : X → Y beameasurable transformation,whichmeansthat
Hereafter
ispreimageof A under T .(Tosimplifythenotation,wewrite T 1 A and Tx rather than T 1 (A) and T (x),respectively,ifitdoesnotcauseconfusion.)
Introduceasetfunction
T HEOREM 1.1.–(Aboutinducedmeasure)Thesetoffunction ν givenin[1.3]isa measureon F .
P ROOF.–Thefunction ν iswelldefineddueto[1.1].Wehavetoshowthatitisnot identicaltoinfinity,butitisnon-negativeandsigma-additive.
Indeed, ν (∅)= μ(T 1 ∅)= μ(∅)=0,andtherefore, ν isnotidenticaltoinfinity. Foreach A ∈ F , ν (A) ≥ 0,because μ isanon-negativesetfunction.
Gaussian Measures in Hilbert Space: Construction and Properties, First Edition Alexander Kukush © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.
Finally, {An ,n ≥ 1} aredisjointsetsfrom F .Thenthepreimages {T 1 An , n ≥ 1} aredisjointaswell,and
Here,weusedthesigma-additivityof μ.Thus, ν isanon-negativeandsigmaadditivesetfunctiononthesigma-algebra F ,i.e. ν isameasureon F .
D EFINITION 1.1.– Thesetfunction ν givenin [1.3] iscalledameasureinducedby transformation T andisdenotedas μT 1 .
Thenotationpromptshowtoevaluate ν (A): (μT 1 )(A)= μ(T 1 A),A ∈ F.
Foranymeasurablespace (Y,F,ν ),denote L(F,ν ) thespaceofLebesgue integrablefunctionson Y w.r.t.measure ν .
Let f : Y → R bean F -measurablefunction,i.e.foreachBorelsubset B of extendedrealline R,itholds f 1 B ∈ F .
T HEOREM 1.2.–(Thechangeofvariablesformula)Assumethateither f ≥ 0 or f ∈ L(Y,μT 1 ).Thenitholds X
P ROOF.–Equality[1.4]isshowninastandardway:firstforindicators,thenforsimple non-negativefunctions,thenfor f ≥ 0,andfinally,for f ∈ L(Y,μT 1 )
a)Let A ∈ F , f (y )= IA (y )= 1, if y ∈ A 0, otherwise.
Then
IA (Tx)= 1, if Tx ∈ A 0, otherwise,
IA (Tx)= 1, if x ∈ T 1 A 0, otherwise,
IA (Tx)= IT 1 A (x)
Hence
IA (Tx)dμ(x)= X IT 1 A (x)dμ(x)= μ(T 1 A)=(μT 1 )(A), Y IA (y )d(μT 1 )(y )=(μT 1 )(A), and[1.4]followsfortheindicatorfunction.
b)Let f ≥ 0 beasimple F -measurablefunction.Thenitadmitsarepresentation
(y )= m k =1 a
,y ∈ Y,
withdisjointmeasurablesets {Ak ,k =1,...,m} andnon-negative ak .Forthe function[1.5],relation[1.4]followsduetopart(a)oftheproofandthelinearityof theLebesgueintegral.
c)Let f beanarbitrarynon-negativeand F -measurablefunction.Thenthere existsasequence {pn (y ),n ≥ 1,y ∈ Y } ofnon-negative,simpleand F -measurable functionssuchthat pn convergesto f pointwiseand pn (y ) ≤ pn+1 (y ), n ≥ 1, y ∈ Y
Bypart(b)oftheproof,
Here,tend n toinfinity.Bythemonotoneconvergencetheorem,[1.6]implies[1.4].
d)Finally,let f ∈ L(Y,μT 1 ), f+ (y ):=max{f (y ), 0},f (y ):= min{f (y ), 0},y ∈ Y.
Bypart(c)oftheproof,
Subtract[1.8]from[1.7]andobtain[1.4]usingthedefinitionofLebesgueintegral.
Problems1.1
1)Let λT 2 beaLebesguemeasureon [0,T ]2 ; π1 (x1 ,x2 )= x1 , (x1 ,x2 ) ∈ [0,T ]2 , π1 :[0,T ]2 → R.Showthat (
where λ1 isLebesguemeasureon R and S1 issigma-algebraofLebesguemeasurable setson R.
2)Let μ1 and μ2 befinitemeasuresonBorelsigma-algebra B (R),and π (x1 ,x2 )= x1 , (x1 ,x2 ) ∈ R2 .Findtheinducedmeasure (μ1 × μ2 )π 1 1 .
3)Fortheobjectsoftheorem1.1,provethefollowing:if μT 1 issigma-finite, then μ issigma-finiteaswell.Doestheoppositeholdtrue?
4)Let f : Y → R beany F -measurablefunction.ShowthattheLebesgueintegral ontheleft-handsideof[1.4]iswelldefinedif,andonlyif,theintegralontherighthandsideof[1.4]iswelldefined.Moreover,incasewheretheyarewelldefined,they coincide.
1.2.InvarianceofLebesguemeasure
Considerameasurespace (X,S,μ) andameasurabletransformation T : X → X
D EFINITION 1.2.– Themeasure μ iscalledinvariantunder T ,or T -invariant,if μT 1 = μ.
R EMARK 1.1.–Assumeadditionallythat T isabijectionon X ,andmoreover T 1 is ameasurabletransformationaswell.Then μ is T -invariantifandonlyif
(Hereafter TB denotesimageof B under T .)
P ROOF.–
a)Let μ be T -invariantand B ∈ S .Because T 1 ismeasurable, A := TB ∈ S Itholds B = T 1 A,and μ(T 1 A)= μ(A).Equality[1.9]follows.
b)Conversely,assume[1.9]andtakeany A ∈ S .Denote B0 = T 1 A, B0 ∈ S . Then (μT 1 )(A)= μ(B0 )= μ(TB0 )= μ(A),and μ is T -invariant.
E XAMPLE 1.1.–(Countingmeasure)Let X = {1, 2,...,n}, S =2X bethesigmaalgebraofallsubsetsof X ,and μ bethecountingmeasureon X ,i.e. μ(A)= |A|, A ∈ S .(Hereafter |A| isnumberofpointsinaset A;if A isinfinite, |A| =+∞.)Then
μ isinvariantunderanybijection π on X .Indeed, μ(π 1 A)= |π 1 A| = |A| = μ(A), A ∈ S
Inthissection,weshowthatLebesguemeasure λn on Rn isrotationand translationinvariant.Hereafter,wesupposethatEuclideanspace Rn consistsof columnvectors x =(x1 ,...,xn ) .
D EFINITION 1.3.– Affinetransformationof Rn isamappingofaform Tx = Lx + c, with L ∈ Rn×n and c ∈ Rn .Suchtransformationiscallednon-singularif L is non-singular.Otherwise,if det L =0,then T ofthisformiscalledasingularaffine transformation.
R EMARK 1.2.–Affinetransformation Tx = Lx + c isinvertibleif,andonlyif,itis non-singular.Inthiscase,theinversetransformationisanon-singularaffine transformationaswell,anditactsasfollows:
T 1 y = L 1 y L 1 c,y ∈ Rn
Rememberthatnon-singularaffinetransformationsonaplaneincluderotations, translationsandaxialsymmetries.
T HEOREM 1.3.–(TransformationofLebesguemeasureatBorelsets)Consider Lebesguemeasure λn onBorelsigma-algebra B (Rn ).Let Tx = Lx + c bea non-singularaffinetransformationon Rn .Then
λn T 1 = 1 | det L| λn . [1.10]
P ROOF.–Transformation T iscontinuous,and,therefore,Borelmeasurable.Hence theinducedmeasure λn T 1 on B (Rn ) iswelldefined.
a)For a =(ak )n 1 and b =(bk )n 1 with ak <bk , k =1,...,n denote [a,b]= n k =1 [ak ,bk ], (a,b]= n k =1 (ak ,bk ].
Hereafter, n k =1 Ak standsforCartesianproductof A1 ,...,An .Evaluate
λn T 1 ([a,b])= λn T 1 [a,b] = T 1 [a,b] dλn = T 1 [a,b] dx.
ThelatterintegralisRiemannintegraloverthecompactandJordanmeasurableset T 1 [a,b].ThechangeofvariablesintheRiemannintegralleadstothefollowing:
λn T 1 [a,b] = [a,b] ∂y ∂x 1 dy = mn ([a,b]) | det L| = λn ([a,b]) | det L| .
Here mn isJordanmeasureon Rn
b)Consideraset (a,b] introducedin[1.11],andlet {ak (m),m ≥ 1} bea decreasingsequencethatconvergesto ak suchthat ak (m) <bk , m ≥ 1; k =1,...,n Denote a(m)=(ak (m))n k =1 ∈ Rn .Then Am :=[a(m),b] isamonotonesequenceof setsthatconvergesto (a,b].ThecontinuityofLebesguemeasurefrombelowimplies λn T 1 ((a,b])=lim m→∞ λn (T 1 Am )=lim m→∞ λn (Am ) | det L| = λn ((a,b]) | det L| .
Here,weusedpart(a)oftheproof.
c)Thus,thetwomeasures λn T 1 and λn | det L| in[1.10]coincideonthesemiring Pn thatconsistsofallbricks (a,b] from[1.11].Bothmeasuresaresigma-finite,and therefore,theycoincideon σr (Pn )= B (Rn ),where σr (Pn ) issigma-ringgenerated by Pn .
Now,weextendtheorem1.3toLebesguemeasure λn onsigma-algebra Sn of Lebesguemeasurablesetsin Rn .
L EMMA 1.1.–Non-singularaffinetransformation Tx = Lx + c is (Sn ,Sn )-measurable,i.e.forany A ∈ Sn , T 1 A ∈ Sn aswell.
P ROOF.–Itisknown(see[HAL13])that Sn = {B ∪ N |B ∈B (Rn ),N ⊂ N0 with N0 ∈B (Rn ),λn (N0 )=0}.
Let A ∈ Sn ,then A = B ∪ N ,with B and N describedin[1.12].Itholds
Here,weusedtheorem1.3andthefactthat T isaBorelfunction.Decompositions [1.13]and[1.14]showthat T 1 A ∈ Sn .
T HEOREM 1.4.–(TransformationofLebesguemeasure)Let Tx = Lx + c bea non-singularaffinetransformationon Rn .ForLebesguemeasure λn on Sn ,itholds
λn T 1 = λn | det L| .
P ROOF.–Consider A ∈ Sn anddecompose T 1 A asin[1.13]and[1.14].Because λn (N )= λn (T 1 N )=0, wehavebytheorem1.3:
λn T 1 A = λn T 1 B = λn (B ) | det L| = λn (A) | det L| .
