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Palm theory, mass transports and ergodic theory for group-stationary processes [Elektronische Ressource] / von Daniel Sebastian Gentner

142 pages
Palm Theory, Mass-Transports and Ergodic TheoryforGroup-Stationary ProcessesZur Erlangung des akademischen Grades einesDoktors der Naturwissenschaftenvon der Fakultat fur Mathematik des¨ ¨Karlsruher Instituts fur¨ Technologie (KIT)genehmigteDissertationvonDipl.-Math. M.Sc. Daniel Sebastian Gentneraus KarlsruheTag der mundlic¨ hen Prufung:¨ 16. Februar 2011Erster Gutachter: Prof. Dr. Gun¨ ter LastZweiter Gutachter: apl. Prof. Dr. Daniel HugiiiiiGratefully dedicated to my parentsClaudia and Hermannand to my dear sisterAnja.ivvPrefaceThis PhD thesis has been written during my employment as research and teachingassistant at the Institute for Stochastics at Karlsruhe Institute of Technology (KIT). Iwish to thank several people at this place for various kinds of support.SpecialthanksgotomyadvisorProf.Dr.Gun¨ terLastforintroducingmetotherealmsofrandommeasuresandStochasticGeometry,forvaluableinputandhintsformyresearchthroughoutthesethreeyearsandhisconstantreadinesstodiscussmymathproblems. Mysecond reviewer, apl. Prof. Dr. Daniel Hug, not only found a number of typing errors ina first version of this text (the remaining errors are all mine) but also provided valuablecommentsandhelp,especiallyconcerningquestionsfromconvexgeometry. Ialsobenefitedfrom discussions with PD Dr. Dieter Kadelka, PD Dr. Stefan Kuhnlein,¨ HDoz. Dr. OliverBaues, Dr. Sebastian Grensing and Dr. Panayotis Mertikopoulos. Dr.
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secondreviewer,apl.Prof.Dr.DanielHug,notonlyfoundanumberoftypingerrorsin
afirstversionofthistext(theremainingerrorsareallmine)butalsoprovidedvaluable
commentsandhelp,especiallyconcerningquestionsfromconvexgeometry.Ialsobenefited
fromdiscussionswithPDDr.DieterKadelka,PDDr.StefanKuhnlein,HDoz.Dr.Oliver
Baues,regularlyDr.calledSebastianmyattenGrensingtiontoandnewDr.boPoksanayonotisseveralMertikdifferenopoulos.tareasDr.ofMartinFmathematicsolkers
andfromIwhicwishhtoIbthankenefitedhimforregularlycreating.IwouldandmainliketotainingthankanallexcellenmembtersofthemathematicalInstitutelibrary
ofStochastics,inparticularMichaelaRegelinandTatjanaDominic,forcreating
friendlyandpleasantworkingenvironment.
hisMysincereextraordinarythanksbogookstoandProf.Dr.throughOlasevveralKallenpapbergerswhothattaugharetinmemanytimatelythingslinkedthroughtopartsof
fromthisIITthesis.Madras.HeEqually,taughIwtouldmelikqueueingetotheorythankwhenProf.IhadDr.PtheanamalaipleasuretoRamaraoTAsomePofarthasarath
thecourseshegaveinKarlsruheandprovidedconstantencouragementandsupport.
IowehugethankstomyfriendsinGermany,GreeceandtheUSaswellastomy
girlfriend.TheyearatBrownUniversityinProvidence,RI,trulyenrichedmylifeboth
personallyandscientificallyandIgratefullymemorizeBrownasanawesomeplacetolearn
mathematics.doandFinally,Iamdeeplyindebtedtomyfamily,inparticulartomyparentsandtomy
sister.Withouttheirrecognition,theirsupportofmystudiesandtheirencouragement
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2.1.1Regularitypropertiesandu-measurabilit.....y........5..
2.1.2Kernelsandtheirregularity.prop..erties...........7...
2.22.2.1HaarandHaarinvmeasureariantandmomeasures..dular....function..............................98........
2.2.32.2.2PropGrouperopoperationserations..and..in.v...ariance...............................1110.........
2.32.2.4DisinNotionstegration....related...to..inv...ariance...of......measures....................1712.........
2.3.1Kernelsandinvariance.....................17.....
2.3.2Disintegrationonproduct..spaces..............17....
2.3.3Invariantdisintegrationofjointlyinv.arian..t..measures...18.
2.3.4Disintegrationofkernelswithinvariance...prop...erties...20.
2.42.4.1RandomRandommeasuresmeasuresandandPstationaritalm..ypairs.................................2222.......
2.4.2Thecanonicalframeworkfor..stationarit...y........27..

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3.1Inversionkernel...........................29.......
3.1.1Constructionoftheinversionk.ernel..............29...
3.1.2Specialoperationsandrespectiveinversion..k..ernels.....31.
3.2Consequencesandapplications....................32.....
3.2.1Disintegration.revisited...................33.....
3.2.2Disprovingproperness.....................34.....
3.2.3Projectingfunctionsfromgroupstoother...spaces......35..
3.2.4Transformingstationaryrandommeasures............38...
3.2.5InvariantPalmk.ernels....................39.....

.(TPMadyFSTP4.1The4.1.1cumulativConstructionePalmofthemeasure.cum..ulativ...e.P.alm....measure....................4242.......
4.1.2BasicpropertiesofthecumulativePalm..measure.......45..
4.2CumulativePalmmeasureofCoxprocesses...............49...
4.2.1SomeclassicalresultsforPoisson.pro..cesses.........50..
4.2.2Coxprocesses.........................51......
4.3Cum4.2.3ulativPePartiallyalmProbabilitstationaryyCox.measurespro....cesses............................54.52......
4.3.1CumulativePalmprobability.measures.............54...
4.3.2ConditionalcumulativePalm.measures.............56...
4.4PropertiesofcumulativePalm.measures...............57....
4.4.1TheTransportTheorem....................57.....
4.4.34.4.2TransportCharacterizationpropertiesofofcumthecumulativeulativPalmeP.alm.measures...measure............5961...

124.........................caseSubspace-stationary7.4.2123...........................caseGrid-stationary7.4.1123......................theoremsdicergotheofApplications7.4122...............eleton-skkrandomaofmeasureytensitIn7.3.2120.................setsBorelofximationapprobiasedUn7.3.1120....................MTPtegratedintheusingApplications7.3115.................................Examples7.2.3113...........groupisometrytheirandmanifoldsRiemannian7.2.2111..................0-cellsandypicalteenwetbRelations7.2.1110................................partitionsRandom7.2105...............tessellationsyx-DelaunaCoofcellsypicalT7.1.4104...................tessellationsyDelaunaandoronoiV7.1.3101........tessellationafromedderivmeasuresalmPeulativCum7.1.2100...............................essellationsT7.1.199...............................tessellationsRandom7.199GMaTMfdyPFefiIIEfaiVLMSeTaVMieaTMCV82..............graphsdularnon-unimoossiblyperansitivT5.4.276............graphsequasi-transitivinsubgraphsStationary5.4.175.....................subgraphsStationaryapplication:An5.474.....................PrincipleortranspMass-TalmP5.3.273..........measuresrandomstationarytoappliedMTPThe5.3.173....................measuresrandomstationaryforMTP’s5.371............................ersionvtegratedIn5.2.270................estativrepresenorbitofsystemsonMTP5.2.170.....................(MTP)PrincipleortranspMass-TThe5.270...........................caseenon-transitivA5.1.367....................casedularnon-unimoetransitivA5.1.266......................casedularunimoetransitivA5.1.165....................................ationsMotiv5.110PdiVIibSMadfdFVeb9FeefiTTPMviii

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6.1.1Anergodictheoremfor.lattice-actions.............86...
6.1.2Sampleintensityforgrid-stationaryrandom...measures....88.
6.2Partiallystationaryrandom.measures................89....
6.2.1Wiener’sergodictheoremandfurther..preparations.......89.
6.2.2Sampleintensit..y.....................94......
6.3ErgodicityandCumulativePalm.measure...............95...
6.3.1Thegrid-stationary..case..................96.....
6.3.2Thesubspace-stationary..case................97....

)5UMfjRhEkjbVdRMdPPkOjVedjheIdInthefocusofthisthesisisthegroupanalysisstationaryofspatialstochastic.processes
Theseprocessesmayconvenientlybedescribedasgroupstationaryrandommeasuresonan
appropriatespace,wheregroupstationarityreferstoadistributionalinvarianceproperty
withrespecttoagroupactingonthisspace.Thispropertyrepresentsarequirement,
whichisweakenoughtoallowtheconstructionofreasonableandwell-fittingmodelsfor
alargevarietyofnaturalreal-worldphenomenasuchase.g.cellgrowthprocesses,the
adevcallelopmentcenofterforestorthefires,developmenrain-droptofmocracdels,kstheinarrivcertainalandmaterials.handlingAtofthephonesamecallstime,init
ofrepresenthesetsmoadels.requiremenHere,rtaandomstrongmeasurenougheistoallonothingwabutareasonablerandomelementmathematicalintheinvspaceestigation
inofallteresttomeasuresus(inonatrocertainductionstofixedthespacesubandjectmathisybe14ric,15foundh,28in,class29[]).ofobRandomjectswillbeofma
measuresrapidlyaresinceinthen.thefoItcusallofmanstartedyresearcwithherstheinspsinceectionthe1950sofinandtegerthevfieldalueddevrandomelopedmeasures
cen(soter.calledpConradointprPocalm,esses)aonSwtheedishreallineengineer,asawasmodeltheforfirstthewhoinvarrivingestigatedphonesuccallshainmoadel.callA
centralobjectfortheexaminationofstationaryrandommeasureshasbeennamedafter
him-thePalmmeasure.Ithasbeendefinedsofarforrandommeasureslivingona
spaceonwhichacertaingrouptractsansitivelyandwithrespecttowhichthemeasureis
stationary.Thetransitivitytogetherwiththecstationaritompleteyenforcestatisticala
spatialhomogeneityoftheprocess.Looselyspeaking,nomatterinwhichpointofspacean
observerdecidestomeasuretherandommassconfigurationaroundhim,hewillneverbe
abletotellfromhis(repeated)measurementswherehesits(nomatterwhicharbitrarily
sophisticatedstatisticaltoolboxheemploys).
TheanalysisofrandommeasuresRondthatarestationarywithrespecttothegroup
ofalltranslations(i.e.withRdrespitselfectto)isbynowaclassicaldomainofrandom
measuretheoryandbecameanindispensablepillarfortheStorealmchasticofGeome-
try(see64[,66]forcomprehensiveintroductions).dStationaryparticlek-flatprocesses,
processes,clusterprocesses,randompartitionsorRtessellationshaveunderonthesta-
tionarityassumptionaspatialhomogeneityproperty.Thispropertyallowsinspiteofthe
factthattheyconsistrealizationwiseofadiscretesetofinfinitelymanyobjects
(whicmakesitimpossibletoaverageovertheseobjectsinthenaivesenseinsuchawaythateach
obthejectcollectionreceivesoftheobsamejectsweighthet),processtheconsistsextractionof.ofArandommeaningfulelementthatdistributionsisassodistributedciated
Fortheaccordingabovetoprosuchcesses,aonespdistributioneaksoftheisintypicdistributionterpretedalobtypicjectasalofpaofa,articletheofarelevtypicantalpro,flatcess.
atypicalclusteroratypical.cellThederivationofallthesedistributionsistheresultof

jorhto

tifi-oftothethetran-tdifferenehsucafrom,ahsuctheanjectsobspacey2Chapter1:IntroductionandOutline
aandmoreoverthesophisticatedspaceonwhicfairhathevproeragingcesslivproes.cedure,bothovertheunderlyingprobabilit
Allthesedistributionsmaybeinterpretedasdistributionsofsuitablerandom
underthePalmmeasure,whichexplainsthecentralroleofthismeasure.Itallows
elegantandunifiedtreatmentofalltheabovementioneddistributions.Meckewas
firstwhousedPalmmethodsinStochasticGeometryinhisseminal47].paperKallen[berg
conarebasedtributedand30in][thatimpareortantalreadyresultson21publishedwhic].Wheinhasome[veofmarktheedcenthetralrelevanresultstintheorems.thisthesis
SomeofthesewerealsoindependentlyfoundbyKallenb31erg].in[
Thisbringsustothefirstmainresultofthisthesis,namelythederivationof
RPd)almwhicmeasurehareevenforstationaryprocesseswithresp(livingectontosomearbitraryabstract,possiblyspace,possiblynon-transitivdifferene,tgroupfrom
actions.PThealmopmeasureerating(whicgrouphneedswenotshallevenbecallunimoforgodular.odAsreasonsmencumulativetionedtheabPalmove,measurthis)ewillgeneralized
helpi.e.ustowhereidenthetifytunderlyingypicalgroupobjectsactionevenisfornotprotransitivcesses,e.thatThisareisnotnotonlycompletelyinterestingstationary
theoreticalviewpoint.Manyreal-worldphenomenaexhibitspatialinhomogeneities,
thattheassumptionofacompletespatialhomogeneityisuntenable.Ourcumulative
Palmmeasurewillallowthemathematicalexplorationofmoreadequate,non-transitiv
models.Thereadermaythinkofe.g.amaterialconsistingofdifferentlayerswith
erties.propAministicsecondmainprinciple,partmass-trofthethisansportthesisprincipleis.Itdevotedconstitutestotheaextensionmass-conservofanationimportanlawtdeter-
forcertaintransportationrules.Again,thisprinciplehasbeenfoundforspecial
thesitivpeossiblygroupactions24non-transitiv,6,[7]eandandphasbossiblyeensubstannon-unimotiallydularcase.extendedAswineour21shall]paptoersho[w,itmay
belinkintoterpretedrandomasmeasureanidentheory,tityandbetconweentributescumulativavePaluablealminmeasurestuitivewhichunderstandingestablishesfor
transformationofonecumulativePalmmeasureintoanotherone.
Thethirdmaintoolthatweshalldevelopfortwospecialtypesofgroupstationary
processeswheretheunderlyingoperationinnon-transitive,areergodictheorems.Wenote
here,thatMeijering49]seems[tobethefirstwhoinvestigatedarandomdgeometricmodel
underergodicityassumptions.Thetwoclassesare,first,randomRmeasuresthatonare
stationarywithrespecttotheoperationofadiscreteZgrid,d,andidensecondtifiedby
dimensionalthecasewherewedhavestationaritywithrespecttotheactionofafixedlower
underlinearergosubspacedicitRy.ofAsitassumptionswillinturntheout,limittheofcumLulativcertainp-conevPa.s.almergenceandmeasureresults,naturallywhicharises
againshowstherelevanceofthecumulativePalmmeasureandestablishesthelink
.theoryalmPWestationaryfinallyshorandomwhoprowthiscesses,whereextendedthetoolboxunderlyingmaybegroupusedforactiontheisinspnotectionnecessarilyofgrouptran-
sitive.Thisincludesaresultongraphautomorphism-stationaryrandomsubgraphs
ercolationquasi-transitive,possiblynon-unimodulardeterministicgraphs(e.g.abondp
withmodel)aswellasrandompartitionsonorientableRiemannianmanifoldsstationary
idenrespecttothenaturalactionoftherespectiveisometrygroup.Italsoincludesthe
cationandstructuralanalysistypicalofCoxDelaunay.cAellsCox-Delaunaytessellation
isaspecialrandomtessellationRd,whereoftherandomnessstemsfromaCoxpoint
process(whichisarandomizedPoissonprocess).
Thethesisisorganizedasfollows:
InChapter2weprovidethereaderwithnecessarybackgroundinmeasuretheory

3andtegrations.alsopresentThissomecrecenhaptertdevalreadyelopmentscontainsconcerningsomethenewresults.existenceMostofinvnotablyarian,wetderivdisin-ea
fromtecsomehnicallymeasurableelaboratespacetoresultaproonductthespace,existenceofexhibitinginvarianatkcertainernelsindisinvariancetegratingpropertany.yk
WethenproceedinChapter3withtheconstructionofanimportantkernel,which
isassociatedtoany‘well-behaving’groupaction.inversionThiskernelwillleadusto
theconstructionofthe‘general’Palmmeasureforarbitrarygroupcumulativeactions,the
Palmmeasure.ItisderivedbyfactoringouttheHaarmeasureoftheoperatinggroup
fromThiscumanotherulativemeasurePalmwhicmeasurehisisannaturallyinterestingassoobciatedjecttoofthestudyrandomonitsomeasurewnofrighint,andterest.we
shallChapterderiv5econsometainsimpourortantextensiontheoremsofaroundthethismass-transpmeasureortin4.principleChaptertonon-transitive
Ingroupparticular,actions,wasewshoellwasthatanimpthereortanistacloseconsequencetrlinkanspbforetortweentherandomoraemformeasurescumandulativtranspeorts.
Palsoalmgivmeasureseafirstandaversionapplicationoftheofthismass-transpprincipleorttoprincipleforrandomautomorphism-stationarytransports.Wesubgraphsof
quasi-transitive,possiblynon-unimodulargraph.
Ourcongrid-stationaritvergenceyresultsandaretheconsubspace-stationarittentof6.yChapterAsmenseparatelytioned,andabovweeweshowtreatintheendofthat
chapterhowthecumulativePalmmeasurenaturallyarisesinthelimitunderergodicit
assumptions.ThefinalChapter7willillustratetheusefulnessofourdevelopedtools.Wegive
structurallyquiteexplicitformulasforthedistributionoftypicalCox-Delaunaycells,
dtheunderlyingCoxprocessisassumedtobestationarywithrespectdtoasubspaceof
R.Thisclearlyincludesthecompletelystationarycase,whereRtheitself,subspaceis
andeventhisspecialcaseseemstobenew.Wealsointroducerandompartitionson
aRiemannianrelationbetwmanifoldseentheseandobjectssuitablywhicdefinehmatheybetypicnotionalofparaphrasedcelandl0-cbyel.ltheWeinderivetuitivelyappealing
ell-knownstatemenfortthatcompletelythe0-cellstationaryisavolume-wtessellationsReighd.Weintedvalsoersionofillustratethettheypicaluseofcell,ourafactw
ergostationarydictheoremsrandombytessellationsderivingRdsomeandofwequiteillustrategeneralconvtheseergencebygivingresultsforsomegrid-examples.orsubspace-
TheseexamplesincludeforRinstance-stationarytessellationsontheinfinitecylinder
R×S1(wheretheactionRonofR×S1isunderstoodtoaffectthefirstcomponentonly
translation).viaeachMorechapterdetailedaswellasreferencesthroughouttorelevtheantthesis.literaturemaybefoundintheintroductionsto

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).(Lybdenotedis)oflawthecalled(alsoSspacemeasurableaintelemenrandomaofdistributionthe)C,A,(ΩspaceyprobabilitaenGivspace.engivaonmeasuresallofspacetheonrelationalenceequivantsrepresentlyevidenandmeasureseenwetbyuittinconabsoluteutualmdenotesrelationThe.νtoectrespwithuoustinconabsolutelyisµthatmeansν≪µrelationtheSonmeasuresareνandµIf).E(σybdenotedebwillEtainingcon-algebraσsmallesttheSof)S(P⊂Esubsetsofsystemaengivand)S(PybdenotedisSsetaofseterwopThe.Son)f·A((ν7→Ameasuretheforν·fwriteoftenew+S∈forF.Sonfunctionsalued]-v∞,[0-measurableSofspacethe+Sybdenoteewspacemeasurableais)S,S(erwhenevurther,Fell-defined.wisiterwhenev)f(νfνybdνf∫tegralinthedenoteew]∞,∞[→S:ffunctionmeasurableaandSonνmeasureaorF.T
Sybdenotedebysaalwwill-algebraσductprotheirTandS-algebrasσowtenGiv.RandT,S-algebrasσeectivrespwithspacesmeasurabledenoteysaalwshallRandT,Sthesis.thisthroughoutusednotationbasicourfixalsoeWsection.thisinernelskandymeasurabilitersalunivofconceptstherecallmainlyshalleWjURehocRMikhRShecdejVediGecR-.)].31[ergbKallenand]21[LastandtnerGenybconstructedtlyendenindeptlyrecenerations,opgroupenon-transitivenevforkernelinversiontheofofproexistenceourintingredieneykaebwillresultthis3ChapterIn].21[LastandtnerGenybestablishedaswThisspace.ductproaonmeasuretarianvintlyjoinaisitselferbmemernelkheacwhereernelskoftegrationsdisintarianvinofexistencethe2.3SectioninincludeTheseresults.newsometpresenalsoewlinestheAlong.2.4Sectioninmeasuresrandomofconceptthefinallyand2.3Sectioninspacesductproonmeasurestarianvintlyjoinoftegrationsdisintarianvinoftheorytheintselopmendevtrecensome,2.2SectioninmeasurestarianvingroupandmeasureHaarofnotionthe,2.1SectionintheorymeasureofnotionsancedadvmoresomeasellwastaryelemensomeareThesethesis.thisinusedconceptsbasicsomewithreaderthefamiliarizetotryewhaptercductorytrointhisIn5UMfjRh-

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Inmanysituationsonehastorequireaµmeasuretobeσ-finite,i.e.toaskforthe
toexistencerequireofitatobs-finiteemeasurable,relaxingB1,partitionB2the,..ab.othatvesplitsconditionitbinytoonlyfiniterequiringpieces,theoratexistenceleast

6Chapter2:Fundamentalsandrecentdevelopmentsinmeasuretheory
µof(Aa)"µsequence(A),Aµn(∈)Sof.finitemeasuresthatapproximatesitsetwisefrombelowsuchthat
nTheconceptofs-finitenesswasintroducedbyKallen30]bwhoerg[notedthatitsimpli-
fiesmanyarguments,mostlysincetheclassofs-finitemeasuresisclosedunderprojections:
ans-finitemeasureMonaproductS×Tinducess-finitemeasuresM(·×T)andM(S×·).
Notethattheanaloguestatementwiths-finitenessσreplaced-finitenessbyiswrong:the
2-dimensionalLebesguemeasureλ2onR2isσ-finite,unlikeitsproλ2(·jection×R).In
additions-finitenessisnotonlypreservedundermostbasicoperationsthatalsopreserv
σ-finiteness-itissometimeseveneasiertoverifythanσap-finitenessossibleproperty.
FinallymostcomputationalrulessuchasFubini’sTheoremarestillvalidfors-finite
sures.Therearemanyreasonsthatmakebothoftheseregularityconceptsnecessarybuttwo
First,particularlyaimpmeasureµortan≠0tononessomearespacetheSfolloisσwing-finitetwo.ifandonlyifthereisastrictly
pofmositivutualeabsolutefunctionf:Scon→tin(0,∞euit)ysucquivalenththatfiniteµf<∞measure.νThis:=f·µgiveswhicrisehmatoyanclearlyinthebesense
assumedtheoreticaltobeapropertiesprobabilitthatyallomeasure.wusThentomostidentifyspacespthemossesswithtopBorelologicalR.subsetsorofmeasurethe
realsbSucetwheenspacesmeasurablearecalledspacesSandBorelTisspaces.calledFBororelaapreciseisomorphismifdefinition,bfothandfa:fS1→areTbijectivemap
measurable(i.e.theBorelisomorphismsaretheisomorphismsinthecategoryof
mea-surablespaces).NowBoraelspaceisameasurablespaceBorelisomorphictoaBorel
subsetofR.Henceσ-finitemeasuresonBorelspacesmayforsomepurposesbetreatedas
probabilitymeasuresR,onwherewemayusetheirdistributionfunctionstoinvestigate
theirproperties.Asaparticularlyimportantexamplewherethisobservationbears
wementionKallenberg’s30[]elegantproofconcerningtheexistenceofdisintegrations
σ-finitemeasuresonproductspaceswherethesecondfactorisBorel.Itwillbeessen
tialforourSecond,indiscussionstegrationinofSubsectionuniv2.3.2.ersallymeasurablefunctionsthatarenotmeasurable
withrespimatedecttobythefinitegivσenmeasures.-algebraLetisusquicmeaningfulklyreponlyeatifthetheinconcepttegratingofunivmeasureersallycanbemeasurableappro
µofsetsSandwithrespfunctionsectµ.tohere.TheµisIfauniversalmeasureconSompletion(,S)ofathenσwe-algebradenoteSisbySthenthedefinedcompletionas
Su=\Sµ
µonewheremaytheequivinalentlytersectiontakeisthetakclassenoveroftheallclassfiniteofSallmeasures,S)onhere).(probabilitS,SThe)yelemen(clearlymeasurestson(
ofuniversalthisσly-algebrameasurare(ableu-measurcalleduniversalableinlymeshort)asurifablesetsSuit/Tandisamap-measurable.f:S→TNotingisthatcalledwe
maydecomposeanysetA∈SuforanarbitraryfiniteµmeasureintoA=B∪CwithB∈S
andC⊂N∈Swithµ(N)=0,itisnaturaltoµ(Adefine):=µ(B).Thisuwaµy(A)is
clearlywell-definedandthusthisdefinitionyieldsanaturalµtoS.extenInationnextof
step,wemayapproximateagivenu-measurableffunction:S→[0,∞]byu-measurable
stepfunctionswhichdefinestheinµfintegraltheobviousway.Fromintegrationwith
respstep:ectjusttonotefinitethatmeasuresanyσsuctoh-finitemeasureorµevmayenbemorewrittengenerallyasas-finitecountablemeasuressumofisafinitesmall
measureswhichyieldsthedesiredextension.
ofRinRecallthethatcategoryaofmeasurablemeasurableBorspaceelbyisspaces,definitionwhereifitisisomorphismsisomorphicaretoabijectionsBorelbsubsetetw

emea-fruitsforx-een

pieces,finitetoinsµthesplitsultaneouslysimhwhicTof...,2B,1Bpartitionaisthereif-finiteσuniformlyandTonmeasure-finiteσais)·s,(µmeasuretheS∈sheacforif-finiteσointwisepisµernelkA.µ"nµwithnµernelskfiniteofsequenceaisthereifs-finite,TtoSfromµernelkacalleW.S∈s,∞<)T(sµiffiniteand,S∈s,1=)T(sµifchasticstoorMarkoviancalledisµernelkA.TtoSfromu-kernelcalledismeasurable’ersally‘univybreplacedis‘measurable’where(2)yertpropdifiedmotheand(1)isfyingsat-]∞,[0→T×S:µmapaSimilarly).·s,(µofinstead)·(sµwritesometimeseW.T∈Afixedyanformapmeasurableais)A,·(µ(2),S∈sfixedyanfor)T,T(onmeasureais)·s,(µ(1)ertiespropthewith]∞,[0→T×S:µmapaisTtoSfromkernelaspaces,measurableare)T,T(and)S,S(Ifwing:follotheistheorymeasureandyprobabilitforjectobtalfundamenARhjVRifhefohRTkbMhVjjURVhMdPKRhdRbi-.).-.f=fgfhence-thsucallfort=)t(gfimplieshwhic),s(f=tthatnoteand)t(g:=sput)S(f∈tforHence).S(f∈t,)fGraph(∈)t,)t(g(thathsucS→T:gfunctionu-measurableayields(ii)2.1TheoremSectiontheBorel,isTSince(ii)).S(f=))f(Graph(Tprthatnotetoremainsitand)fGraph(ofjection-proTtheto(i)2.1TheoremjectionProtheapplyymaewHence).t,)s(f(=)ts,(,2T→T×S:under2T⊂}T∈t:)tt,({diagonal(measurable)theofpreimagetheisitsinceT×SofsubsetmeasurableaisIt.}S∈s:))s(fs,({:=)fGraph(settheisfofgraphThe(i)2.5].Lemma,31[fromentakareofsprowingfolloTheof.oPr.f=fgfsatisfyingS→T:gfunctionableasuru-meaiseTher(ii)able;asuru-meisfof)S(fimageThe(i)able.asurmeebT→S:fandesacspelBorebTandSetLerse).vineakwandrange().)8MTTFfurther.enevy-measurabilituoftagesanadvthetshighlighlemmanextThe392].p.,18[or252]p.,16[Seeof.oPr.)A(Spr∈slalforA∈))s(fs,(thatsuchT→S:ffunctionableasuru-meaiseTher(ii)able.asuru-meis}T∈tsomeforA∈)ts,(:S∈s{:=ASprsetthei.e.,SonAofctionojeprThe(i).T
S∈AandeacspelBoraTe,acspableasurmeaebSetLsections).andjectionspro().(TPMadMTosal:dispourattheoremsctionseandctionojeprerfulwopevhaewsetsandfunctionsu-measurableorF13.17]).10.15,8.15,atzeS,59[e.g.(seeHausdorffandtablesecond-councompact,callyloisologytopwhosespaceologicaltopyantainingconenevclassugehaagainishwhicA.47])Theorem,11[andA1.2]Theorem,28[(seeBorelisspaceseparable)andmetrizablecompletely(i.e.olishPyaninstanceorFspaces:ofclassugehaisthisevoabindicatedAsdirections.othbinmeasurablearethatspacesmeasurable72.1Somenotionsfrommeasuretheory

.2.12and2.9,2.8,2.6LemmainresultsnewaccessibleklyquicsometpresenalsowilleWerations.opgroupundersetsandmeasuresofariancevinandmeasureHaargroups,ologicaltopconcerningfactsbasicsomesummarizeewsectionthisIncRMikhRijMhVMdlVdMdPBMMh-.-µ(s,Bis)=(i+1)(i+11≤f(s,t)<1ii+11µ(s,dt)
{∫≤(i+1)(i+11≤f(s,t)<1if(s,t)µ(s,dt)
{∫≤(i+1)µsf(s,·)<∞.
(ii)criterionUseσthefor-finitesamemeasuresconstructionforevaserya)insfixed∈(whicpartS).(hboilsdowntoprovingthewell-known

BPr1so,Bof.2s,.(i).of.TLetµbsucehaσthat-finites,t()7→k({ernelt∈Bis}fromStoisT,i.e.measurableforsforeac∈ianS∈hyNthereandµsis(Bisa)<∞.partition
sThenwedefine∑11
0<f(s,t):=i>01+µ(s,Bis)2i({t∈Bi}≤1
thatnoteandsµsf(s,·)=1+µ(µs,(s,BiB)s)12i<1.
∑i0i>Conversely,fixf>0onS×Twithµsf(s,·)<∞.Define
11Bis:=t∈T:i+1≤f(s,t)<i,i∈N,
{andnotethatindeed<sincef<0∞thisdefinesapartitionTforofeverys∈S.In
addition

8Chapter2:Fundamentalsandrecentdevelopmentsinmeasuretheory

Si.e.toµ(Ts,Bσi)<-finite∞,ifs∈Sfor,i∈seacN∈.hSInbthereetweenisathesetwomeasurableconcepts,Bwspartitione,Bs,call..aof.Tµksucernelfromh
21thats,(t)7→({t∈Bis}ismeasurablefori∈allNandµ(s,Bis)<∞,s∈S.Clearly,a
uniformlyσ-finitekernelσis-finiteandanysuchkernelinturnispoinσt-finite.wiseJust
asthereisafunctionalcσharacterization-finitenessofofmeasureswehavethefollowing
functionalcharacterizationsofthelattertwotypesofkernels.
8MTTF).-(regularitypropertiesofkLeternels).µbeakernelfrSomtoT.Thefol-
holds:lowing(i)µisσ-finiteithereexistsameasurfable:S×mapT→(0,∞)suchthatµsf(s,·)<
∞,s∈S(inthiscfasemaybechosensuchthatµsf(s,·)<1,s∈S).
(ii)fµ(s,is·)p,s∈Sointwise,arσeme-finiteasuriabletherµesandf(s,exists·)<af∞:,mapSs×∈ST.→(0,∞)suchthatthemaps

2.2Haarandinvariantmeasures9
-.-.)BMMhcRMikhRMdPcePkbMhSkdOjVed
ByGwedenoteagroupwithneutralelemene.tElementsofagroupGwillusually
bge7→gdenoted1arebygorh.measurable,IfGthencarriesweσaGcalla-algebrameGasursuchablethatgroup.themapsgSimilarly,()h7→G,ghifandcarries
atopologyOsuchthattheabovemapsarecontinuouswecalltopoloitgicaalgroup.
Clearly,anytopologicalgroupbecomesameasurableonewhenendowedwiththeBorel-
σ-algebraB(O):=σ(O).Itisclearthatleft-orrighGaret-shiftshomeomorphismson
inthetopologicalandBorelisomorphismsinthemeasurablesetting,suchthattranslates
gA:={ga:a∈A}forg∈GandA∈Gareagainmeasurable.Aλmeasuredefinedon
ifleft-invariantisGλ(gA)=λ(A),g∈G,A∈G,
(right-invariancisedefinedwiththeobviousmodification).Inthetopologicalsetting,a
thenclassicalthereisanresultuptoisconstanthattmwhenevultiplesertheuniquelygroupcarriesdeterminedaloRleft-incallyadonvariancompactmeasuretHausdorfftopology
λ=finite̸0indefinedtheonsenseitsthatBorelσit-algebra.assignsfiniteHerevthealuestoRadoncompactpropouterertyresets,meansgularisonλthatallislocalBorelly
meaningsets,λ(A)=inf{λ(U):U⊃A,Uopen},A∈B(O),
andinnerregularonallopensets,meaning
λ(A)=sup{λ(K):K⊂A,Kcompact},A∈O.
Suchameasureλiscalled(left)aHaarmeasuronethegroupGwithrespecttothe
topandologyharmonicO.Thefolloanalysis.wingAprotheoremofmaybeconstitutesfound20,ina[Theoremscornerstone11.8,of11.9].measuretheoryongroups
dorTPMadMTgroupGp)..(ossessesexistenceanuptoandpositiveuniquenessmultiplesofHaaruniquelyAmeasure).nylocdetermineallydcompHaaractmeHaus-asure.
andItiscrucialtonoteatthispointthattheRadonconditionenforcesanintimate
onimptheortanonethandrelationwithbtheetweennaturaltopologymetricandtopmeasure.ologyAsandgenerated(x,y)example,:=byjxyj,considerx,y∈RR,thereals
andontheotherhandequippedwiththediscretetopologysuchthatA⊂anRyissubset
open.differBothandtopareologiesgivenasare(mlocallyultiplesofcompact)LebandesgueHausdorffmeasurebutinthetherespfirstectivcaseeandHaar(mmeasuresultiplesof)
countingmeasureinthesecond.
setasTheabcorrespoveexampleondingRHaarwithmeasurediscretealsotopshoologywsthatandhenceHaarcounmeasurestingneedσmeasurenot-finiteboneinitspower
general.Toenforceσ-finitenessofaHaarmeasureλitisenough,forinstance,torequire
thetopologyonGtobesecond-countable.Then,thereisacountableGinpartitiontoof
relativelycompactsubsets,oneachofλiswhicfinitehbydefinition.
IfthetopologyonGisassumedtobesecond-countableinaddition,thenonemay
droptheinnerandouterregularityconditionsofHaarmeasurestatedabove-theywillbe
impliedbythelocalfiniteness.InfactmanyauthorsconsiderHaarmeasureandRadon
Inmeasuresthisonlyinthesisλthiswillmorealwspaysecificdenoteσa-finitesecond-counHaartablemeasure28setting,on,p.a36,groupe.g.Gp.,[41].whichin
turntheisexistencealwaysofassumedsucλ.hWetoabewilllocallyabbreviatecompact,theseHausdorffconditionsandbyGsasecond-counisyinglcsc,thattable,thetoensure
Wenow‘Hausdorfffixa’leftconditionHaarmeasurebλeingonaunderstolcscodgroupGto.beTheconleft-intainedvinariancethe‘lopropcallyertymaybcompact’econdition.

,

10Chapter2:Fundamentalsandrecentdevelopmentsinmeasuretheory
rephrasedusingtheusualtandemconsistingofstepfunctionsandmonotoneconvergence
writingybf(hg)λ(dg)=f(g)λ(dg),h∈G,f∈G+.
∫∫Forfixedh∈Gthemeasureλh:A7→λ(Ah)isbyassociativityofgroupmultiplication
areagainleft-inhomeomorphisms).variant,Hence,byclearlyuniquenessnon-zeroofandloHaarcallymeasure,finitethere(noteisthatauniqueGleft-pandositivrighe
isconstancalledt,mowhicthedularhwecallhfunction),(such(thoughthatλλhw=as(h)neededλ.Theinmapthe:G→construction(0,∞),hof7→(h)hereitis
concleartinuousthat(indoesnotparticulardependonthemeasurable)choiceofhomomorphismG→(0λ:normalization,).∞)Itev(seeenofe.g.20,[constitutesProp.a
satisfying11.10])f(gh)λ(dg)=(h1)f(g)λ(dg),h∈G,f∈G+,(2.1)
∫∫andhastheadditionalpropertythat
f(g1)λ(dg)=(g1)f(g)λ(dg),f∈G+.(2.2)
∫∫AgroupGiscalledunimodularifg()=1forallg∈G.By(2.1)Gisunimodularifand
onlyifλisright-invariant.TwoexamplesofclassesofunimodulargroupsaretheAb
eliangroupsandthecompactgroups(whiletheAbeliancaseisimmediateGnoteforcompact
thatcompactthemcontinultiplicativuityandesubgrouphomomorphism,∞)of-(0proptheertonlyyofsuchimplysubgroupthattheevidenG)setmtlyust(b{b1e}einga).
DMTFdR).0.Formanyresultsinthisthesistopologicalrequirementsaresubordinate
inandvwearianσtmigh-finitetaswellmeasureλ=just̸0.GassumeKallentobbeergchomeasurableosesthisandsuchconsequenthattanditmorecarriesnaturalaleft-
topsettingology-freeinvthisarianrecentoft31papthe]ermoand[dularinfunctionparticular2.1)provsatisfyingandes(the2.2)(([31,existenceLemmaof2.3]).ameasurable,
Astheexistenceofleft-inσvarian-finitetmeasuresλ≠0isstillopenwithouttopological
extraassumptionsweshallalwaysconsidertopologicalgroupsinthisthesis.
-.-.-AhekfefRhMjVediMdPVdlMhVMdOR
LetwhicGbhewaegroupabbreviateandSgas:=bset.y'(gAn,ops)er,g∈ationG,sof∈GSon,Sissatisfyinggivenbbesya=oths,map's:∈GS×,S→whereSe,
denotestheneutralelementG,ofandg(hs)=(gh)s,g,h∈G,s∈S.Ifsuchanoperation
alongwiththerelevantmapisgivenwithoutriskofconfusionthenweomitthemapand
simplysaythatGoperatesoractsonS.InthiscasewealsoG,→writeS.
WheneverthegroupGismeasurableσwith-algebraGandthesetSisameasurable
spacecarryingσa-algebraSwenaturallyrequireanoperationGonofStobemeasurable,
inimpliesthesensethethatmeasurabilitthepryunderlyingojeofGthe×ctionsmapπSs→:GS→isSG,
sS∈/SS,and-measurable.shiftsg:S→ThisS,g∈G,condition
ybengivπs(g)=g(s)=gs,g∈G,s∈S.
everWhenevtheerGandunderlyingSaremaptopfromG×ologicalStoSis.spaces,Clearlywe,callcontinanGopuous,→opSerationcerationsontinuousarewhen-measurable
withrespecttoσBorel-fieldsandthecontinuitypassesontoprojectionsandshifts.

t-shiftson

exvconaisSonmeasurestarianv-inG-finiteσallofsetTheRhMjVediefRhPhef-.-..Borel.isSifinstance,forcase,theisThismeasurable.are,S∈t,}t{setstoinponetheifmeasurableareGofsubsetsThese.t=ss,tgsatisfyingtelemengroupfixedaiss,tgwhere,s,sGs,tg=s,tGsinces,sGofcosetleftaotherwiseor)Gs̸∈t(ifyempteitheris,)}t{(s1π=}t=sg:G∈g{:=s,tG,S∈tsecondatakingand,sofstabilizerthedenotes)}s{(s1π=}s=sg:G∈g{:=s,sG,S∈sforurthermore,F.T∈t,S∈sG,∈g,)tgs,g(:=)ts,(gerationopdiagonalthetoectrespwithtarianvinisitif-invariantGjointlyitcallew,T→,GandS→,GerationsopengivwithT×SspaceductproaoneslivµIf.S∈AG,∈g,)A(µ=)Ag(µif,-invariantGSonµmeasureasimilarlyand,G∈gA,=Agif-invariantGcalledisS⊂AsubsetAeration.optheoforbitstheclearlyarejectionsproThese(i).2.1TheoremtoaccordingSinsetu-measurableaisSonsAofjectionprotheBorelisGSince.S∈s),}s{×G(ψ:=sAsetsmeasurabletheand)sg,g(=)s,g(ψybengivS×G→S×G:ψisomorphismBoreltheConsiderof.oPr.Sofsubsetsableasuru-meearorbitstheel,BorisGewherS→,GationeropableasurmeanyGivenorbits).ofyu-measurabilit().18MTTF.ygeneralitlargeinholdshwhicorbits,theofyertpropymeasurabilitwingfollothenotetoterestinginisIt.λmeasureHaar-finiteσwithlcscisGwhereS→,GerationopmeasurableaconsiderwnoeW.sg=twithG∈gaysaalwisthereS∈ts,yanforthatyingsaybrephrasedebymaytransitivitthatNoteorbit.oneonlyossessespandansitivetrcallediserationopthe)S∈sallforhence(andS∈ssomeforGs=SerwhenevandsoforbitthecalledisGs=)G(sπsettheS∈sfixedorFthat2.1])Lemma,30[(seeedwshoalsoHespace.lcscaongrouplcscaofis)S⊂KerwhenevcompactisG⊂)K(s1πthathsucerationopuoustincona(i.e.erationoperpropologicallytopaofnotionclassicalthegeneralizesand]30[inergbKallenybducedtroinaswconceptThis.N∈n,S∈s,∞<)nB(sµthathsucSof...,2B,1BpartitionmeasurableaofexistencetherequiresthisthatrecalleW-finite.σuniformlyenevis,S∈s,s1πλ:=sµardspushforwallofsetthethathsucandmeasurableothbiserationopthethatsensetheinSonerlyoppreratesopGthatassumeewthis,enforceoT-finite.σeb2.2Haarandinvariantmeasures

not11

needtheyeration,optheonassumptionsyregularitfurtherwithoutButjections.prounderλmeasureHaartheof,S∈s,s1πλardspushforwthearemeasurestarianv-inGforexamplesprimeThet.arianv-inGand-finiteσagainisbν+aµmeasurethe0ba,andνµ,hsucyanforthatsensetheincone

12Chapter2:Fundamentalsandrecentdevelopmentsinmeasuretheory
propernessisequivalenttotheexistenceofameasurablek:S→function(0,∞)such
thatµsk=k(t)µs(dt)<∞,s∈S.(2.3)
∫Asindicatedearliertopologicallyproperoperationsareproperinthismore
generalsense.6kFTbSMe.theop(i)erationTisrivially,measurable.compactIfgroupstheyopoperateeratepropconerlytinonanuouslyyonameasurabletopspaceologicalifspace
thentheactionisclearlytopologicallyproper.
(ii)Onereadilydprovesthat≤foru1≤danyu-dimensionallinearLsubspaceofRd
operatesonRviatranslationtopologicallyproper.
(iii)ZdoperatesonRdviatranslationtopologicallyproper.
11(iv)RoptranslationeratesontheinthecylinderfirstZ:=compR×onenS,t.SThisdenotingoperationtheis1-dimensionaltriviallyunittopcircle,ologicallyvia
propd(v)(evTheen)grouppropoferlyonrigidtheanemotionsGGrdoperatesassmannianAtop(d,k),ologicallythespacepropofRerkonallbutacts-dimensionalnot
anesubspacesofRd.Theproblemhereisthatthestabilizerk-dimensionalofa
anecompact.spaceAsEweconwilltainsshoallwintranslationsCorollary3.10,propwithervopectorserationsconE,withtainedi.e.(evincannotenbonlye
locally)closedstabilizersautomaticallyhavecompactstabilizers.
(vi)Givenagraph=V(,E)withcountablesetofvVerticesandedgesetE⊂V×
VitsautomorphismgroupG=Aut(),endowedwithasuitabletopology,acts
topologicallyproperVoninthecanonicalway
Aut()×V→V,(',s)7→'(s).
WepostponethedetailstoSubsection5.4.1.
(vii)GivensuitableatopRiemannianology,actsmanifoldtopM,itsologicallyisometryMpropinergroupGtheon=I(Mcanonical),endowwaedywitha
I(M)×M→M,(',s)7→'(s).
postpThisoned,isinalsothistruecasefortoclosedSubsection7.2.2Gsubgroups.(anofexampleis(iv)).Again,detailsare
-.-.0DejVedihRbMjRPjeVdlMhVMdOReScRMikhRi
ThewhereGcon,v→exSconeisofproper,σallhas-finitea(resp.strikingGs-finite)-instructurevariantthatmeasureshasboneenaBorelilluminatedS,spacebyKallen
in30[,Theorem2.4](resp.31,[Theorem4.2]).Weconsiderσthe-finitecasefirst.As
notedbeforetheprojectionµmeasuress,s∈S,areinvariantmeasuresS.onTheyhave
theadditionalpropertythat
µgs=(g1)µs,g∈G,s∈S,(2.4)

ber.erg

(2.6).O∈b,1=wbµthatsuchSon)∞,(0→S:wfunctionableasuru-meaisetherThen.Oitlalcandonefixmaywethatsuchesentatives,eprrorbitofsystemableasurmeaisetherthatassumeandSonerlyopprateeropGetLfunction)-symmetricGaof(existence).9.8MTTF:wersionvnormalizedwingfollotheybreplacediskwhensimplifycalculationssome)2.3(fromkfunctiontheusingofInstead.yertpropBoreltheinheritsalsospaceBorelaonmeasuress-finiteenevor-finiteσallofspacetheifproblemenopanebtoseemsIt.SfrominheritedishwhicSonmeasuresyprobabilitofspacetheofyertpropBorelcrucialtheandernelksuitableayborbitstheofelinglabtarianvinan(ii),2.2Lemmaonrelies2.4]Theorem,31[inevgahetargumenThefunction).hoicectheofimagetheybengivclearlythenish(whicorbitstheofestativrepresenofsystemafixingpreviouslywithoutfunctionhoicecu-measurableaofexistencetheedvproandassumptions)theseofyanneednotesdotargumenevoabthethat(noteerpropebtoerationoptheandBorelebtoSrequiredinsteadergbKallenBorel.isGsinceu-measurableis)B(1βthat(i)2.1TheoremwithtogetherimpliesthisS∈O,Bforand)))O\B(×G('(Spr=)O\B(G=)B(1βevhaew),sg,g(7→)s,g(,S×G→S×G:'IsomorphismBorel-therecallingHence,).O\B(G=)B(1βevhaewS⊂BforthatNoteof.oPrable.asuru-meisβthen,S∈OIfselectors).ofyu-measurabilit().38MTTFwing:follotheistargumenThe.Oofymeasurabilitrequiretohad]21[LastandtnerGen,βofymeasurabilitestablishtoorderIn.Gsofetativrepresenfixedpreviouslythedenotes)s(βwhereS→S:β)ctorseleorbitcalled]31[erpaprelatederg’sbKallen(infunctionechoictheconsiderymaewandGb,O2b∪=SuniontdisjointhetoinsplitsSspacethe,S∈s,GsorbitstheofestativrepresenofOsystemaosingCho.λofariancev-ins,sGtrighbutnothingmeansthis)2.1(ybthatNote(2.5).S∈s,s,sG∈g,1=)g(enforceserationoptheonupconditionernesspropthethatmeanshwhicProof.SinceG,→Sisproperwemaychooseby30[,Lemma2.1(i)]astrictlypositive
measurablefunctionk:S→(0,∞)suchthatµb(k)<∞,b∈O.Now
(k)sw(s):=µ(s)(k),s∈S,
isu-measurablebyLemma2.8andFubini’stheoremand,inaddition,
µbw=k(s)µb(ds)=1,b∈O.
∫kµbNotethatGifisnotunimodularitwouldnotbeconsistenttoµsw=require1forall
s∈Sby(2.4)whichmakestherestrictionOin(2.6to)necessary.Wenowreformulatea
resultofKallenb30erg,[Theorem2.4].

13

2.2Haarandinvariantmeasures

.bδ)b,bG(λ)}b{(νO2b∑=ν)Otoectresp(withthat)2.7(fromwsfolloit,O∈b,sδ)b,bG(λGb2s∑=sδ)b,sG(λGb2s∑=bµλofariancevleft-inybSincees.tativrepresenorbitofOsystemtable)(counafixandSonνmeasuretarianv-inGaeakT.S∈s,∞<)s,sG(λ<0ifonlyandifcasetheisthisthatstates3.8LemmaOur.SspacetablecountheonerlyproperateopGLet).Stablecoun().((6kFTbSM.2ν=1νhenceand,+S∈hh,2ν=h1νyieldsarbitraryis+S∈hand)2.6(inasiswwhere))s(β(h)s(w:=)s(fputtingand,+S∈f,)db(2ν)ds(bµ)s(f∫∫=)db(1ν)ds(bµ)s(f∫∫Then.Oontratedconcenareand)2.10(satisfyothb2ν,1νosesuppthisseeoT:νybdetermineduniquelyalreadyisνsincekoftendenindepisνdefinitionitsofspiteIn0.=)B(νhasOwithtdisjoineingbS⊂Bmeasurableyanthatsensethein,OontratedconcenSonmeasure-finiteσais1β)ν·k1)k(µ(:=νwhere(2.10),)db(ν)·(bµ∫=)ds(ν)s(k1)k()s(µ)·()s(µ∫=)ds(ν)s(k)·(s'∫=)·(νthat),2.9(and)2.8(usingnote,ew]21[LastandtnerGeninAs.kofhoicectheonenddepnotesdohwhic)2.9(inastationrepresenahsearcwnoeW(2.9).)ds(ν)s(k)·(s'∫=)·(ν2.4])Theorem,30[(cf.aswrittenebymaSonνmeasurehsucyansinceSonmeasurestarianvin-finiteσallofconeexvcontheofgeneratorextremal(normalized)aasusedebcanernelkthisthatedvproergbKallen.Sonernelkais'i.e.theorem,ubini’sFybmeasurableadditioninishwhicorbitstheofelinglabaiss'7→smaptheords,wotherIn(2.8)G.∈g,S∈s,s'=sg'i.e.orbits,ontconstanenevand,S∈s1,=ks'thatsensetheinnormalizeduniformlyt,arianv-inGareS∈s,ks/µsµ:=s'measuresthe)2.3(inaskosingChoof.oPr(2.7).)db(ν)·(bµ∫=)·(νsatisfyingOondateentronccνeasurmeuniqueaisetherSonνeasurme-invariantG-finite,σanyforThentatives.esen-eprrorbitofOsystemableasurmeafixandSeacspelBortheonerlyopprateseropGoseSupperg).bKallenmeasures,tarianvinofositiondecompdicergo().(æTPMadMT(2.11)

14Chapter2:Fundamentalsandrecentdevelopmentsinmeasuretheory

yertpropthetorefers-symmetryGwhere,S∈Bsets-symmetricGofcollectiontheisclasssecondThe.Iybdenoteewhwhic-algebraσaformsetsTheseG.∈gA,=Agmeansariancev-inGwhere,S∈Asets-invariantGtheofconsistsclassfirstThethesis.thisinplaceseralsevatroleecialspayplawillSofsubsetsofclassesowtS→,GerationopanenGiv.2.8LemmaybengivsimplythenisfunctionhoicecciatedassotheS∈OfixingAfterapplications.laterinusefulebtooutturnwillthisandonesmeasurableaccessibleeasilyandexistingytplenusuallyapplicationsintheamongonetenienvconparticularlyaosehoctoyflexibilitfullesgivitthattageanadvthehashapproacOures.tativrepresenofsysteminducedthei.e.),S(βrangetheoutabinformationyanwithoutleftisonewnside,dotheOned.kheccebnotneedsOestativrepresenorbitofsystemmeasurableaofexistencethethatistageanadvancurse:andblessingothbebcanthist,oinviewpapplicationanromF.yawenon-constructivainselectororbitanoftenceexis-theesderivandestativrepresenorbitofsystemmeasurableaofexistencetherequirenotneedshethatistsargumenofstreamhisoffeaturehnicaltecA).3.1.1Subsectioninconstructwillewh(whicernelkersionvinciatedassotheandselectororbitmeasurableu-texistentionedmenevoabtheusingtheoremevoabtheofersionva4.2]Theorem,31[inedvproergbKallenmeasures,tarianv-inGs-finiteofconeexvconthetourningT2.2Haarandinvariantmeasures

DMTFdReFVL6kFTbSMe.(i)ItisevidentthatGan-inyvariantsetisaunionoforbits
andthatanysuchunionGis-invariant.

Proof.Thedecomposition2.7)(implies
ν(A\B)=(A(gb)(B(gb)λ(dg)ν(db),
∫∫andsinceAisG-invariantthismeans
ν(A\B)=(A(b)(B(gb)λ(dg)ν(db)=δ(B)(A(b)ν(db),
∫∫∫whereweusedtheG-symmetryofBinthelaststep.

holds.

δ(B):=µbB(2.12)
whereb∈Oisfixed(andarbitrary).Wenotehereasimplepropertyoftheseobjects.
measur8MTTFeνon).()S(,inavGariant-invariantmeasuresAset⊂onSandsymmetricaGin-symmetrictersections).setBGiven⊂StheaGrelation-invariant
ν(A\B)=ν(A)δ(B)(2.13)

0<µbB=µcB<∞,b,∈cO.
Thelattercollectionisnotevenclosedwith\,∪resporcect.Thetouseofthedefining
propertyofitsmembers,namelythattheyconsistoffiniteandnon-zeropiecesofeachorbit
willbecomeapparentonceweusetheminthefollowingchapters.GGiv-symmetricena
setB⊂Swemaydefineitswidthas

15

of.consistssetthetranslatesoferbumnthejustissetahsucofwidthThe.dZ∈z,d1),[0+ztranslatestegerintheofunionsfiniteallofcollectiontheybengivissets-symmetricGofcollectionThe.2.1Figurewingfollotheindraftedaresets-symmetricGofExamplesradii.arbitrarywithoriginthearoundcirclestricconcentheofyanofunionstdifferenossiblepallybengivaresetstarianv)-ind(OSHere:dR→,)d(OS(iii)mind.insituationsextremetheseeepktolaterapplicationsfortortanimpebwillItt.arianv-inGwnoisS⊂Asubsetyanwhilereasons)hnicaltecforexcludedissetyempt(theSisset-symmetricGonlythecase:etransitivthetocomparedersedrevaresetssymmetric-Gandtarianv-inGofrolestheHereorbit.wnoitsistoinphFigure2.1:ExamplesSofO(2)-symmetricsubsetsBofR2.

eac(iv)L,→RdwhereLisafixedk-dimensionallinearsubspaceRdwhereofk∈{1,...,}:d
L-invariantsetsareunionsofparallelL.translatesExamplesofofL-symmetricsets
aredraftedin2.2Figure.

thatFigure2.2:ExamplesLof-symmetricsubsetsBofR2whereL={(x,0):x∈R}.

sense(v)Zd,→Rd:AnyZd-invariantAsetmayberepresentedbymeansofauniquely
determinedsubsetA0⊂[0,1)dsuchthat
A=A0+Zd.

theinansitivenon-trlytotaliserationopThis.S→,}e{=GwheniscaseextremeotherThe.∞<)B(λ<0butnothingmeansthisleft-translationviaG→,GthatcasetheIn).S∈callforcasethisinthen(andS∈conefor∞<)B(cµ<0ifonlyandif-symmetricGisS∈Bsetaand}S,{;=IthenetransitivisS→,GIferations:opetransitivarefirstThesituations:extremetortanimpowtareThere(ii)theorymeasureintselopmendevtrecenandtalsundamenF2:Chapter16

17tegrationDisin2.3jRThMjVed6ViVd-..pDisineciallyintegrationsStocemergehasticatmanGeometryy.differenTheretareplacesninumerousAnalysis,examples,andProbabilitamongytheseTheoryareandes-
Cavalieri’sprinciple,thecalculusofconditionalexpectationsandprobabilities
ticleclassicalprocessesPalmandmanformalismyotherthatmeaningfulleadstoobthejects.notionInofthise.g.tsectionypicalweobsummarizejectsofthestatestationary
formofknownthetotheartauthor.concerningNewisexistenceLemma2.15ofand(inthevarianelabt)oratedisinexistencetegrationsresultinintheLemmamostgeneral
2.17establishingmeasurablylabeledinvariantdisintegrationsofmeasurablylabeled
joininvariantmeasuresonproductspaces.
-...)KRhdRbiMdPVdlMhVMdOR
IfGoperatesonbothSandTandµisakernelSfromtoTthenµisG-invariantif
µ(gs,gA)=µ(s,A),s∈S,A∈T,g∈G.(2.14)
Thecovarianceproperty
f(t)µ(gs,dt)=f(gt)µ(s,dt),g∈G,f∈T+,
∫∫toisasequiv‘covalentariance’to2.14)(whicorh‘equivistheariance’reasonwhiny‘inthevariance’literature.νofandkGivνernelsenonSismeasuresandsometimesreferred
21TrespectivelytheirproductmeasureS×Tonisdenotedνb1y
ν2.Conversely,givena
measureMonS×T,Misusuallynotaproductmeasure.Still,foralargeclassofsuch
measuresasimilardecomposition,disintecalledgr,ationispossible-eitherStoTfromin
termsofonemeasureνonSandakernelµfromStoTorviceversafromTtoSwhere
themeasurelivesTonandthekernelisTfromtoS.AdisintegrationMfromofStoT
thenreads,givνenandµ,
Mf=f(s,t)µ(s,dt)ν(ds)=:(ν
µ)f,f∈(S
T)+.
∫∫-...-6ViVdjRThMjVededfhePkOjifMORi
InProbabilityTheorydisintegrationsariseforinstancewheneverajointdistribution
tworandomelementsηandisconditionedononevariable.Then
L(η,)=L(η)
C(∈η·j=·)=:L(η)
L(jη=·),
henceνprobabilit:=L(ηy)andmeasureνµand(s,·)a:=MarkC(o∈vianη·j=ks)µernel.isaKallenvbaliderg30[]disinderivedtegrationtheasexistenceaboveofwitha
disintegrationsσ-finiteformeasuresMonproductspacesusingtheprocedurementioned
:2.1.1SubsectioninBorel8MTTFisσ).(..-finitei(disinthereσistegrationaσ-finite-finitemeofasurνeonmeasures)SAmeandasuraσeM-finiteonS×kernelµT,fromwherStoeTTis
thatsuchµ.
ν=MHereµmaybechosenMarkovianM(i·×T)isσ-finite.

andpar-thetlyof

18Chapter2:Fundamentalsandrecentdevelopmentsinmeasuretheory
Proof.gration.See30[The,Lemmaconverse3.1]isformosttheeasilyimplicationseenbyinσvthat-finiteokingaM2.3.LemmaTheadmitslastsuchaassertiondisiniste-
trivialaswemaychoν=oseM(·×T)inthiscase.
Wemayevenextendthisresultslightlytothes-finitecase:notethatanys-finite
measureMonS×TwhereTisBorelmaybedisintegratedbymeansofaprobability
thatmeasureνadmitsonSaandansequences-finiteofkfiniteernelµµknfromernelswithSµnto"Tµ.orHereequivanalens-finitetlymakyberneleiswrittenakasernel
acountablesumoffinitekernels.
8MTTF).(0(disintegrationofs-finiteLetmeasures).Mbeans-finitemeasureSon×
T.ThenM=ν
µforafinitemeasurνeonSandans-finiteµkernelfromStoT.
HereMisσ-finiteµimaybechosenσ-finite.Furthergivenσa-finitemeasurν˜eonS
suchthatM(·×T)≪ν˜thereisasuitables-finiteµ˜frkernelomStoTwithM=˜ν
µ˜.
AsaboveMandµ˜aresimultaneouslyσ-finite.
Proof.ThecaseM=0istrivialsuchthatwemayMassume≠0.IfM=Pn1Mn
withfinitenon-zeromeasuresMnonS×TwehavedisintegrationsMn=νn
µnwith
finitenon-zeromeasuresνnandfinitekernelsµnbyLemma2.14(orsimply,afterobvi-
ousmodifications,theexistenceofconditionaldistributions).Nowdefinethe
probabilitmeasure1ν:=n12nνn(S)νn.
∑Sinceνn≪ν,n∈N,andeachmeasureunderconsiderationisfinite,thereareRadon-
Nikodymdensitiesfn:S→[0,∞)withνn=fn·ν.Usingthese,wemaydefinethes-finite
µ(s,·):=fn(s)µn(s,·).
kernelµfromStoTvia∑
1nThenthemonotoneconvergencetheoremyields
ν
µ=ν
fnµn=(fn·ν)
µn=νn
µn=Mn=M.
∑∑∑∑n1n1n1n1
IfMisσ-finite,thenLemma2.14yieldsadisinMtegration=ν
µwithσ-finiteνand
σ-finiteµ.Usingafunctionf>0onSwithνf<∞wemayrewritethisdisintegration
viaM=ν
µ=(f·ν)
(1fµ)whichyieldsthedesireddisintegration.Theconv
ersetrivial.areimplicationsForthelastassertion,µfixwith˜thestatedpropertiesandconsiderafixeddisinte-
grationM=ν
µ.Hereνandµmaybechosensuchthatµ(s,T)>0,s∈S,since
A:={s∈S:µ(s,T)>0}ismeasurableandwemayform(A·νandredefineµoutside
ofAmeasurablesuitably.functionThenf0onνclearlySM(with·×νT=)≪f·νν˜˜.andPuttingtheµ(s,˜·)Radon-Nik:=f(s)oµ(dyms,·),sTheorem∈S,yieldsyieldsa
ν˜
µ˜=˜ν
(fµ)=(f·˜ν)
µ=ν
µ=M.
-....IdlMhVMdjPViVdjRThMjVedeSWeVdjboVdlMhVMdjcRMikhRi
ofNoteaG-inthatvGifarianopteratesmeasureνononbSothSandandaGT-inandvtheariantdisinkµernelfromMtegrationS=toνT
,µthenMconsistsis

y

19tegrationDisin2.3jointlyG-invariant,inthesenseMthatisinvariantwithrespecttothediagonaloperation
g(s,t):=(gs,g)t,g∈G,s∈S,t∈T,ofGonS×T,i.e.
f(gs,g)tM(d(s,t))=Mf,g∈G,f∈(S
T)+.
∫Conversely,itisnaturaltoσaskif-finite,ajoinGtly-invariantmeasureMonS×Tadmits
suchaninvariantdisintegrwhereationbothνandµareG-invariant.Thisisinfacta
problemthathasbeeninthefocusofmanyauthorssincethe1960’s.Twomainapproaches
weresuccessfulindifferentcontexts:skewthefactorizationapproachofMatthes45][
(1963)andthecombinedregularizationandperfectionapproachwhichappearedfirstin
tlyinthevpaparianert61[]bmeasureyonaproRyll-Nardzewskiductspacein1961.requiresTheoneofclassicalthetwoskewS,factors,tobefactorizationsaytheofajoin
groupmeasureG.MonThenG×theTintobijectivtheeskmeasure#M(g,ew-shift#)ton:=G(g,×g)Ttwhictransformshistheinvjoinariantlytinwithvarianresptect
tosuchshiftsmeasureinistheaprofirstductcomponenmeasuret(only).oftheItλ
isformρthenwithonlyσaa-finitesmallstepmeasureρtoonTdeduce,thatany
andreversingtheskewshiftthengivesthedesireddisintegration.Kallenbergsignifican
thegeneralizedgeneralthissettingforapproac31]hjoinbinytly[shoinvwingarianhotwS×thisTmeasuresbyteconusinghniquethemainyvbeersionevenkappliedernel.in
Wenicelyshallsuppgivorteatheshortrelevancesummaryofofthehisinvideasersioninkernel3.2.1Subsectionforwhictwhoisreasons:partofFirstthisthesethesisandsecond
weshallneedthemtoestablish3.9Theorem(whichseemsnewinthisgeneralityandwill
beneededinthisformlater).
Ontheotherhandtheregularizationandperfectionapproachismoreelaborate:one
firstneedstoidentifyaninvsupparianortingtmeasureνonS,i.e.σa-finiteinvariant
measuresatisfyingM(·×T)≪ν.ThisisdicultsinceitmayhappMen(·×Tthat)is
notthisσas-finite.aspThenecialacase)complicatedwhichcomprehendsconstructionthefollows(seeregularization2.17ourwhicLemmaofhaconfamilytainsofRadon-Niko
invdensitiesariantasone.wGenelltnerasanandavLasteragingusedprothisGceduresmotecovothinghniqueertotheconstructresultingthekinernelvinersiontoankernelin
[21]andweshallpresentthisconstructionin3.1.1.Subsection
andIngavhisesome2007paper30applications[,SectiontoP3]alm(andKallenbergrelated)extendedkernels.andUsingcomparedthebothmethoregularizationdsand
perfectionapproachheproves30]in[thefollowingtheorem:
TPMadMT).(1.(invariantdisinσtegrations-finiteofinvariantmeasures,Kallenberg)
(i)Aσ-finitemeasurMeonS×T,wherTeisBorel,isGjointly-invariantifandonly
ifitadmitsaninvariantdisinteSgrtoT.ationfrom
(ii)IfthenMasthereinis(i)σisa-finitejointlyGandG-invariant-invariantνandMµkernel(fr·×omT)StoisσT-finitewithM=andGν
µ.-invariant
(iii)Ifin(ii)M(·×T)isσ-finite,thenwemaychoνose:=M(·×T)andtheassociated
G-invariantµisstochastic.
Proof.gration(i)mayAbeprooffoundof30inthe,[Corollaryimplication3.6].thatTheGancony-invversejoinariantlyMfollotwsadmitsfromasuchsimpleadisinte-calculation.
(ii)30,is[Theorem3.5]and(iii)istrivial.
Wewillfamiliesalsoofgivjoineatlyinvcompletearianprotofofmeasuresanonproextensionductofspacesthis2.17in.InresultLemmatoadditionmeasurablywelabeled
willfurtherextendthistheoremtothecaseMofins-finiteTheorem3.9.

dymtly

,GandS→,GerationsopandTandS,RspacesmeasurableenGivRhjVRifhefMhVMdORlVdmVjURhdRbiaeSjRThMjVed6ViVd-...0theorymeasureintselopmendevtrecenandtalsundamenF2:Chapter20.R∈r,rκ
rν=rMthatsuch,R∈rG,∈g,S∈s,T∈A,)Ag1s,(rκ=)As,g(rκ,)·,·,r(κ:=rκwritingerty,oppreinvariancthewithTtoS×Romfrκkernel-finiteσaisetherthen,R∈rforrν≪)T×·(rMwithSoneasurme-invariantGais,R∈r,rνachethatsuchStoRomfrkernel-finiteσaisνandR∈rachefor-invariantGjointlyisrMthatsuchisMIf(iii).R∈r,r′κ
r′ν=rMthatsuchTtoS×Romfr′κkernel-finiteσaisetherthen,R∈r,r′ν≪)T×·(rMwithStoRomfrkernel-finiteσais′νIf(ii).R∈r,rκ
rν=rM,)·,·,r(κ:=rκwritingthat,suchTtoS×Romfrκkernel-finiteσaandStoRomfrνkernelchasticstoaiseTher(i).TandSothbonablyasurmeateeropGletandT×StoRomfrkernel-finiteσaMel,BorearTandSewheresacspableasurmeebT,S,RetLernels).koftegrationsdisintarianvin().(28MTTF.3.1.1Subsectioninernelkersionvintheofconstructiontheinoltomainouraseservwillit3.5]Theorem,30[intsargumenerfectionpandregularizationtheand107]p.,28[infoundtsargumenofadaptionardtforwstraighaisofprotheThough].30[inergbKallenybfoundmeasurestarianv-inGtlyjoinforersionsvtarianv-inGeectivrespProof.(i)WemayassumethatMr(S×T)>0,r∈R.SinceMisσ-finitewemaychoose
byLemma2.3ameasurablefunctionf>0onR×S×TsuchthatMrfr=1,r∈R,
andPropdefineositionthesto7.26]chasticyieldsaPkstoernelfromcRκtohasticfromS×RkT×asernelSPtor˜T:=sucfrh·Mr,thatr∈R.togetherThen[28,withthe
stochastickνrernel:=Pr(·×T)
Pr=νr
κ˜r,r∈R,
c.f.Dellacherie/Mey16,er5.58].[Thisisclearlyequivalentto
Mr=νr
κr,r∈R,
whereκ(r,s,A):=∫(A(t)f(r,s,)t1κ˜(r,s,dt),A∈T,andthusprovesthefirstassertion.
νfrom(ii)abν′ovIfiseasatisfiesgivνenkMernel(·R×fromTto)≪Sν′,withr∈Rthe,andpropbyertMyr(Dellac·×T)≪νr′herie/Mey,r∈R16,,er[then5.58]
rrrrwemaychooseameasurablefunctionf:R×S→[0,∞]suchthat
f(r,)s=dνdνrr′(s),rν′-a.e.s∈S.

theirand6.3])Theorem,28[e.g.(seespacesductproonmeasuresoftegrationsdisinofexistencetheonresultswnknoofextensioncrucialaishwhiclemmawingfollotheneedewthisorF.yawmeasurableandtarianvinaninT×SonR2r}rM{measurestarianv-inGtlyjoinoffamilieseledlabmeasurablyofositiondecomptarianvinandmeasurableaevprotoissubsectionthisinaimour,T→

ybThen(2.18).R∈r,S∈s,,srκ)s(LA(:=,srκ˜defineewFinally0.=)rcA(rνthatimplies)2.15(andtarianv-inGisrAthatkhecceasilycanoneurtherF.ymeasurabilitdesiredtheealsrevhwhic,}0>)s,r(Bz2CBsup{(1=)s(LA(writeymaewHencezero.are)s,r(BztermstheC∈BheacforiffrA∈sevhaclearlyewurtherFTheorem.ubini’sFybmeasurableis))qp,(d(2λ˜∞}<)qp,,r(Bm{(j)B)(qs,qrκp,psrκ(j)q(l)p(l∫∫:=)s,r(BzalsoThenG.∈qp,,R∈r,})B(qs,qrκ,)B(p,psrκ{min:=)qp,,r(BmmapmeasurabletheC∈BheacfordefineandTofCclassdeterminingmeasuretablecounaetakthisorFmeasurable.is)s(LA(7→)s,r(mapthethatwshownoeW(2.17).R∈r,rA∈sG,∈h,h,hsrκ=,srκevhaew,R∈r,}2G∈)qp,-a.e.(2λ˜,qs,qrκ=p,psrκ:S∈s{:=rAsetstheonthatwssho3.5]Theorem,30[inascalculationsimilarA(2.16).)dh)(λ˜·l)(h,hsrκ(∫:=,srκdefineymaewThen1.=λl˜withGonfunctionmeasurablesomeeb0>lLet(2.15).R∈r,S∈s-a.e.rνG,∈g-a.e.λ˜,g1,srκ=s,grκparticularinyieldstheoremubini’sFGonλ˜measureHaartrighsomeFixing.R∈rG,∈g,S∈s-a.e.rν,g1,srκ=s,grκesgivhwhicclassdeterminingmeasuretablecounaparticularinadmitsitBorelisTSince.R∈rG,∈g,)ds(rν)dt(g1,srκ)ts,(f∫∫=)ds(rν)dt(s,grκ)ts,(f∫∫that+)T
S(∈fyanforimplyrνandrMofariancevIn.R∈r,rκ
rν=rMwithTtoS×Rfromκernelkagetew(ii)romF(iii)t.statemensecondtheesvprohwhic),·s,,r(κ)s,r(f:=)·s,,r(′κwhere,R∈r,r′κ
r′ν=rMThenernel.ktheofctionerfepcalledis)2.18(inersbmemernelk‘nice’ofselectionthewhilegularizationerastoreferredis)21

2.16Disin2.3tegration

(inκernelktheofothingsmotheHere).(3.DMTFdRindeed.hold,R∈r,rκ˜
rν=rMtegrationsdisinrequiredthe,S∈s-a.e.rν,,srκ=,srκ=,srκ˜sinceand,R∈r,T∈A,S∈sG,∈g,)A1g(,srκ˜=)A(s,grκ˜)2.17(andrAofariancevin

22Chapter2:Fundamentalsandrecentdevelopmentsinmeasuretheory
-.0FMdPeccRMikhRiMdPijMjVedMhVjo
InframewSubsectionorkfollo2.4.1wewingfirstKallendefineb31erg]randominand[measurespresentinsomeaofverytheirgeneralprop(non-toperties.Then,ological)in
Subsection2.4.2wedefineanddiscussgroupstationarityofrandomelements.
-.0.)FMdPeccRMikhRiMdPPMbcfMVhi
Letendo(wS9,S)(Sb)eawiththemeasurablesmallestσspace-field9Mand(S(S))therenderingspaceoftheσall-finitemappingsµ7→measuresµ(B)onS.forWeall
B∈notationsSevenmeasurable.thoughCLetneed,(ΩA,notC)bbeeaaσ-finiteprobabilitmeasureymeasure.space.InWeuseparticular,probabilisticwedenote
ξby:EΩin→9(Stegration)thatσwithis-finiterespC.ectinAtortheandomsensemethatasureonforωS∈eacisΩhatheremeasurableisacounmappingtable
partitionB1ω,B2ω,..of.Ssuchthatξ(ω,Biω)<∞C-a.e.ω∈Ωforanyi∈Nandsuch
thatω,(s)7→({s∈Bω}ismeasurablei∈forN,i.e.ξisnothingbutσa-finitekernelfrom
ΩtoSusingthekiernelnotationξ(ω,B):=ξ(ω)(B).ApointprocessonSisarandom
measureonSwhichchargesallmeasurablesetswithN∪{v0,alues∞}.inWedenotethe
identitymaponΩbyeinordertobeconsistentwith2.26).(
IfξisarandommeasureonSthentheCampbellmeasureCξofξwithrespectCtois
themeasureonΩ×Ssatisfying∫
Cξf=Ef(e,s)ξ(ds),f∈(A
S)+.(2.19)
Furtherηifisarandomelement∫inaBorelTspacethen
Cξ,f=Ef(η,s)ξ(ds),f∈(T
S)+,(2.20)
iscalledCampbtheellmeasureofthep(airξ,η).Thesemeasureshavethefollowing
erties.prop8MTTF).(9(propertiesofCampbellmeasures).(i)Givenarandommeasurξeona
measur(ii)ableGivenspracSeandomitsmeCampbasurelesξlmeandξ˜asuronCeξaisBorσelsp-finite.aceS,then
ξ=ξ˜C-a.e.,Cξ=Cξ~.
(iii)GiveninadditionarandomηinelementaBorelspacTetheCampbellmeasure
ofthepair(ξ,)ηiss-finiteandσitis-finitewheneverξisη-measurableEorξisσ-finite.
Proof.(i)isevidentinviewof2.3.Lemma
(ii)Oneimplicationoftheequivalenceistrivial.ToCseeξ=theCξ~,other,suppose
i.e.Cξf=Cξ~f,f∈(A
S)+.
SinceSisBorelthereisacountablemeasuredeterminingC⊂S.FclassorB∈Cthe
specialcf(hoiceω,s)=g(ω)(B(s)foranarbitraryg∈A+yieldsξ(B)=ξ˜(B)C-a.e.,and
sinceCiscountablethisyields
ξ(B)=ξ˜(B),B∈C,C-a.e..
AsCismeasuredeterminingthisyieldstheassertion.
(iii)Thes-finitenessis30pro,vedLemmain[4.2]aswellasσthe-finitenessinthecase
whenfunctionξisfη>0on-measurable.SsuchthatTheEξfcase<∞Esucwhenξhisσthat-finite∫alsof(iss)Cξ,(d(immediatet,s))=Easξfw<e∞ma.ychosea

2.4Randommeasuresandstationarity

23

Thedefinedσvia-finitenessEξ()(A)Cof:=ξEdoξes(A)not,A∈S.necessarilyNotethatEcarryξ=oCverξ(Ωto×·)theintensityandthatmeσasurEeξ-finitenessofξ
isusuallynotpreservedunderprojections.EξButiss-finite,sincetheclassofs-finite
measuressometimesisthecloseduseofundersuppproortingmejectionsasures-aofξkeynecessaryobserv.ationTheseofσareKallen30-finite].bergThisinmeasures[makes
equivalentEtoξinthesenseofmutualabsolutecontinuity.
8MTTF).)æ(s-finitenessanditsuse,Kallenberg).
(i)Anys-finitemeasureonaproductspacS×eThass-finiteprojectionsM(·×T)and
M(S×·).
(ii)GivenarandommeasurξeitsintensitymeasurEξeiss-finite.
F(iii)oranys-finitemeasurνethereisafinitemeν˜asureν.
(iv)Anyrandommeasureξpossessesafinitesupportingmeasure.

MPrno(of.S×·(i))MareIfn"MapproisanximatingapproximatingsequencesofsequencefiniteofmeasuresfiniteM(·×forTmeasures)Mandn(M·then×(TS)×·and),
respectively.Now(ii)folloCwsξisσsince-finite,inparticularEs-finite,ξ=Cξand(Ω×
·).For(iii)wetakeasequenceoffinitenon-zeroνn"νmeasuresandnotethatν˜:=
Pn2nνn(S)1νnhasthedesiredproperty.Now(iv)followsfrom(ii)and(iii).
NotethatthedefinitionEξoftogetherwiththemonotoneconvergencetheoremyields
Ef(s)ξ(ds)=f(s)Eξ(ds),f∈S+.
∫∫ThisidentitywillbeusedfrequentlyandCampbisell’scalledTheoreminsomepartsofthe
literature.IfΩisBorelthen2.14Lemmayieldsσa-finitemeasureνonSandaσ-finite
kernelQfromStoΩdisinCtegratingξasfollows:
Cξf=f(ω,s)Qs(dω)ν(ds),f∈(A
S)+.(2.21)
∫∫Wecallanypairν,(Q)satisfying2.21)(aPalmpairofξ(see21[]).ThekernelQisthe
ν-associatedPalmkernelofξ.Tomakethedependenceonξexplicit,wesometimeswrite
(νξ,Qξ):=(ν,Q).QmaybechosentobestochasticifandEξonlyisσif-finiteinwhich
caseSincethestructuralmeasuresQsarerequiremenprobabilittsonΩyaremeasuresnotonΩ,desirablePalmtheprobonemayabilityconsidermeasuronΩ.esinsteadarandom
withelementηtheinafactCBorelξ,thatisspaceTalwandaysforms-finitesimilarto2.21according)(bymeansto2.19ofLemmaLemma(iii))2.15adisin(togethertegrationof
formtheCξ,f=f(t,s)Ps(dt)ν(ds),f∈(A
S)+.(2.22)
∫∫Givenadisintegrationas2.22)inwe(calltheP-kernelmembersν-associatedPalm
(pseudo)distributionsPs,s∈S,ofη.Wheneveraν-associatedPalmkQernelexists
clearlythenPs=Qs(η∈)·ν-a.e.
ItstocishasticclearexpfromerimenthetQsthanwhicconstructionhPsmakconesthatittainslesssometimesinformationpreferableontotheworkunderlyingdirectlyon

.nSonesscoprointpais)n(ξtheness,coprointpaisξandelBorisSIf(iv).nSoneasurmeandomraisnξ(iii).T×Soneasurmeandomrais)ds,ω(ξ)dts,,ω(γ·}∈)ts,({(∫∫:=)·,ω)(γ
ξ((ii),Soneasurmeandomrais)ds,ω(ξ)s,ω(f·}∈s{(∫:=)·,ω(η(i)Then.TtoS×Ωomfrkernel-finiteσaγandfunctionableasurmea)∞,[0→S×Ω:f,SoneasurmeandomradenoteξetLmeasures).randomoftransformations().)(8MTTFhere.lemmahnicaltecationmenfinallyeWely).ectivrespnorder(ofξofesasurmemomentfactorialthecalledaremeasuresytensitinTheirts.onencomptdifferenpairwisewithnN∈)ni...,,1i(allervoentakissumthewhere,elyectivrespnSonN∈n,)nξ,...,)ξ(δ=6)n,...,i)i(∑:=)n(ξmeasuresrandomtheformymaewcaseeitherIn.Sinitselemenrandomofsequenceinfinite)or(finiteondingcorrespawithitselfNervoor,N∈nsomefor}n,...,1{formtheofsetfiniteaervoeitherentakissumthewhereξδiP=ξwriteymaewSspaceBoreltheoncessprotoinpaisξifaddition,In.ξtocteesprwithηofs)nξ·jj∈η(CdistributionsPalmderth-ornthe-finite,σisnξEthatassumptiontheunderyields,ξ
···
ξ=nξmeasureductprorandom-foldnitsybξ)2.22(inReplacingΩ.ofertiespropstructuralonenddepnotesdoexistencewhoseand4.1.1SectioninducetroinwillewthatO×ΩonEmeasurecertainaebwillatoklotojectobetterbThespace.yprobabilitabstractunderlyingouronconditionsyregularithsucyanoseimpnotshallewandBorelisΩifinsuredonlyisexistenceTheir(pseudo)-distributions).almPthetimessome-only(andthesisthisinpairsalmPuserarelywillew]21[orkwourtotrastconIn(2.24).+)S
)S(M(∈f,)ds)(ξE(s)ξjjdµ∈ξ(C)sµ,(f∫∫=)ds(ξ)s,ξ(f∫Ereads)2.22(andξE=νosehocymaewthenspace,Borelainaluesv-a.s.Cestakξand-finiteσisξEifAgain,true.is)S(9fortstatemenondingcorrespthenotorwhetherproblemenopanebtoseemsIt564].561,pp.,28[inastsargumensimilarwingfolloybisSerwhenevBorelis)S(P9thatwnshoebcanIt).S(9ofsubsetmeasurableais(2.23)}N∈i,∞<)iB(µ:)S(9∈µ{:=)S(P924Chapter2:Fundamentalsandrecentdevelopmentsinmeasuretheory

ΩinsteadonT,seeSubsection2.4.2.IfEξisσ-finitesuchQthatsandPsareEξ-a.e.
stochasticPsthenisalsocalledPalmthedistributionηofwithrespectξtoats∈Sand
weshallwrite,followingKallen32,33b],erg[inthiscase

C(η∈·jjξ)s:=Ps(·),s∈S.
Ofparticularimportanceistheη=caseξ.whenIfthereisafixedSpartitionintoof
measurablesetsP:={B1,B2,.}..suchthatξ(Bi)<∞C-a.e.thenξ∈9P(S)C-a.s.
where

2.4Randommeasuresandstationarity

25

Proof.(i)Weneedtoshowηthatisσa-finitekernelfromΩS.toChooseh:Ω×S→
(0,∞)suchthatξ(ω,h(ω,·))<∞andletA:={(ω,s):f(ω,s)>0}.Thenput
h(ω,s)
g(ω,s):=(Aβ(ω,s)+(A(ω,s)f(ω,s)>0,ω∈Ω,s∈S,
andobservethatηg(ω)=({(ω,s)∈A}h(ω,s)ξ(ω,ds)<∞.Theassertionnowfollows
∫(i).2.3Lemmafrom(ii)Againweneedtocheckσmeasurable-finiteness.Choosef:Ω×S×T→(0,∞)
withδ(ω,s,f(ω,s,·))<1,ω∈Ω,s∈S,andfξ:Ω×S→(0,∞)withξ(ω,ξf(ω,·))<
1,ω∈Ω.Puttingf:=ffξtheassertionfollowsagainfrom2.3Lemma(i).
ξ(ω,Ffor(ω,·))<(iii)1andchodefineosegby:Ω2.3Lemma×Sn(i)→a(0,∞)bmeasurableyffunction:Ω×S→(0,∞)suchthat
g(ω,s1,...n),s:=f(ω,s1)...(ωf,sn)
thenclearlyξn(ω,g(ω,·))<1.TheassertionnowfollowsfromLemma2.3(i).
(iv)nowfollowsfrom(iii)ξ(n)(ω,sincef(ω,·))≤ξn(ω,f(ω,·))togetherwiththefact
thatξ(n)takesvaluesNin∪{0,∞}.
ThenextlemmaisduetoMeck46e[].
8MTTF).))(tamingofrandommeasures).Letξbearandommeasureonthemeasur-
ablespacSe.
(i)Thereisafunctionh:Ω×S→(0,∞)suchthat
h(ω,s)ξ(ω,ds)=({ξ(ω)≠0},ω∈Ω.
∫(ii)Ifξisuniformlyσ-finitewithrespecttoaP,partitionthereisahfunction:9P(S)×
S→(0,∞)suchthat
h(ξ(ω),s)ξ(ω,ds)=({ξ(ω)≠0},ω∈Ω.
∫Proof.(i)ChoosebymeansofLemma2.3ameasurablef:Ω×S→(0,∞)suchthat
ξ(ω,f(ω,·))<∞.Then
f(ω,s)
h(ω,s):=({ξ(ω)≠0}∫f(ω,t)ξ(ω,dt)+({ξ(ω)=0},ω∈Ω,s∈S,
hasthedesiredproperty.
(ii)P=If{B1,B2,.}..thenwemayput
1Pia(µ,s)=21+µ(Bi)(Bξ(s),µ∈9(S),s∈S,
∑iandnotethatµif≠0wehave
0<a(µ,s)µ(ds)<1.
∫Wemaythendefine
)sµ,(ah(µ,s):=({µ≠0}∫a(µ,)tµ(dt)+({µ=0},µ∈9P(S),s∈S,
whichhasthedesiredproperties.

.)ds(nη)s,nη(h)ηj)µt,(d∈)η,ξ((C)sµ,t,(f∫∫E=)ds(nη)s,nη(h)s,η,ξ(f∫E)∞,[0→nS×)S(P9×T:fmeasurableyanforyields)S(P9×TspaceBorelthein)η,ξ(:=ξ˜ariablevrandomtheto)2.25(Applying).2.25(edvproewusthand,)ds(nη)s,nη(h]ηjC∈s,B∈ξ[C∫E=)ds(nη)s,nη(h)ηjB∈ξ(C}C∈s{(∫E=)ds(nη)s,nη(h}C∈s,B∈ξ{(∫EviaceedproymaeW.)ds(nη)s,nη(h}C∈s{(∫]ηjB∈ξ[C[E=)dµ∈η(C)ds(nµ)s,nµ(h}C∈s{(∫)µ=ηjB∈ξ(C∫=)dµ∈η(C)µ=ηjdt∈ξ(C)ds(nµ)s,nµ(h}C∈s{(∫}B∈t{(∫∫=)ds(nη)s,nη(h}C∈s,B∈ξ{(∫Ethatηonconditioningybobtainewleft,Starting.nS∈CandT∈BwhereC×B=Awhencasethetoreducefurthersame),thetiallyessenishwhic1.17]Lemma,28[ashsucresultuniquenessa(ortargumenclassmonotoneayb,ymaew,Ainmeasures(!)finitearesidesothbAs.nS
T∈Aallfor,)ds(nη)s,nη(h)ηjdt∈ξ(C}A∈)st,({(∫∫E=)ds(nη)s,nη(h}A∈)s,ξ({(∫Ethatwshotoenoughisitergence,vconmonotoneybandfinmeasuresaresidesBoth(2.25).)ds(nη)s,nη(h)ηjdt∈ξ(C)st,(f∫∫E=)ds(nη)s,nη(h)s,ξ(f∫E+)nS
T(∈fyanforthatwshofirsteW.Ω∈ω,}0≠)ω(η{(=}0≠)ω(nη{(=)ds,ω(nη)s,)ω(nη(h∫with)∞,(0→nS×)nS(P9:hfunctiona(ii)2.22Lemmaofmeansybosehocew,nSofPpartitionthetoectrespwithysa,nηmeasurerandomthetoervocarriesclearlyηof-finitenessσuniformtheAsof.oPr.ηgivenξofdistributiononditionalctheofversiongulareradenotes]η·j∈ξ[Cewher,+)nS
T(∈f,)ds(nη)ηjdt∈ξ(C)st,(f∫∫E=)ds(nη)s,ξ(f∫EN∈nanyforThen.Seacspableasurmeaonηeasurmeandomr-finiteσuniformlyadenoteηletandTeacspelBorainelementdoman-radenoteξetLmeasure).tegratinginantoectrespwithconditioning().)-8MTTF.7Chapterinmosaicsx-DelauneyCoofdiscussionourforneededebwillsubsectionthisinlemmalastThetheorymeasureintselopmendevtrecenandtalsundamenF2:Chapter26

2.4Randommeasuresandstationarity27
finallyhereosingCho˜f(t,µ,)s=hf(µ(nt,,s)s),t∈T,µ∈9P(S),s∈S,
forarbitraryf˜∈(T
S)+yields
Ef(ξ,η,)sηn(ds)=Ehf(µ(nt,,s)s)C((ξ,)η∈d(t,µ)jη)h(ηn,s)ηn(ds)
∫∫∫=Ef(t,s)C((ξ,)η∈d(t,µ)jη)ηn(ds)
∫∫=Ef(t,s)C(ξ∈dtjη)ηn(ds),
∫∫whichistheassertion.
-.0.-HUROMdedVOMbShMcRmehaSehijMjVedMhVjo
respect(Partial)toanoperatingStationarityofagroup.randomTomeasurecapturerefersthistoinvdescriptionarianceofpreciselyits,considerdistributionanopwitheration
G,→actionSG,of→9some(S)lcscofGongrouptheGonspacesomeofσallmeasurable-finitespacemeasuresS.onSThisviaoperationinducesan
gµ(·):=µg1(·)=µ(g1·),g∈G,µ∈9(S).
Thereasonfordefiningtheshiftofameasureintermsgof1arathershiftbythanashift
bygisthatthischoiceleadstothecovarianceproperty
f(s)(gµ)(ds)=f(gs)µ(ds),g∈G,f∈S+,
∫∫bymeasureusingwhosemonotonelawLcon(ξ)visGergence.a-invNoarianw,atGprobabilit-stationaryyrandom9measure(Smeasure)onwithξisrespaectrandomtothe
inducedoperationG,→9(S),i.e.
C(ξ∈gA)=C(ξ∈A),g∈G,A∈M(S).
g∈EvidenG.tlythisStationaritisyequivisaalenpurelyttosagyingξdistributionalhasthatthesamepropertyanddistributionisξindepforasendeneacthoftheconcrete
functionalrepresentationξasofarandomelementof9(S).Still,amongthemany
possibleparticularlyrepresenusefultationsandξconasofvaenienmaptfromforPaalmspaceΩcalculus:to9(S)ChothereosingistheonethatcanonicalissettingΩ:=
9(S)theidentityξmap(ω):=ωbecomesarandommeasurewithdistributionC.Hence,
G-stationaritξyisofnothingGbut-invarianceCinofthissetting.Evidentlytherelation
ξ(gω)=gω=gξ(ω)
holdsinaddition.Hencestationaritymay(withoutanylossofgenerality)berepresented
bythefollowingmathematicalframework.
AssumethatGoperatesmeasurablyonΩ(wedonotneedtorequireanyfurther
regularitsistencies:ytheconditionsopGeration,→here9(S)andisinfarfactfrome.g.properassumingforpropnon-compacternessG)andherewriteleadstoheavyincon-
gωguish:=gbω.etwTheeengroupreasonelemenfortsthisthemselvsuddenescandhangeactualofnotationshiftsonisΩ,thatwhicithbenablesecomeΩ-vustoalueddistin-

t.arianv-inGebtoCosehocandG,∈g,Ω∈ω,))ω(n,...,)ω(1(g=))ωg(n,...,)ωg(1(thathsucΩosehocymaewAgain,e.livtselemenrandomthesewherespaces,theofductprotheonGoferationopdiagonalthetoectrespwithdistributiontjointheirofariancevinasdefinedisn...,,1tselemenrandomeralsevofe-invariancGJointt.arianv-inGebtoCandsatisfiesthathsucΩosehocymaewevoabastsargumensimilarBymeasure.tarianv-inGais)(Lif-stationaryGisthen,TintelemenrandomaisandTspaceaonmeasurablyeratesopGIfts.elemenrandomofystationaritdefineymaewmeasuresrandomofystationarittoSimilarlycess.prostationaryaofjectsobypical’‘tedderivmoreorowtatokinglowhenformalitiessimplifiesThisailable.vaisterimenexphasticcstotheoutabinformationthelalwhereellevtheatenhapptheythattage,anadvugehtheevhaitselfΩonshiftsthebuttationrepresenfunctionalahsuctooneselfrestricttomeanssettingcanonicaltheosingChoΩ.spaceecificspaoftermsintationrepresenfunctionalthenotusuallyterest,inofistelemenrandomaofdistributiontheonlytheoryyprobabilitIntessellation.thisoferticesvtheofcessprotoinptheis)ω(ξwhile,dRoftessellationsofspacetheinelivtmighωinstanceorFlost.ebtmighinformationsince)ω(ξybnotbutωybcapturedelyentirishwhict’erimenexphasticcsto‘underlyingthisofatederivaisformedthenismeasurealmPthehwhictoectrespwithξmeasurerandomThemeasure.randomorcessproparticletessellation,set,randomstationaryae.g.t,elemenrandomcomplicatedquite-stationaryGausuallyisterimenexphasticcstounderlyingtheTheoryalmPInork.wframe-evoabtheoftagesanadvthethighlightoadequateseemordswadditionalSome.esasurmeandomrinvariantmeasuresrandomstationarycallauthorssomeywhreasontheishwhicernel,ktarianv-inGabutnothingframethisinismeasurerandom-stationaryGathatNote(2.27).S∈B,Ω∈ωG,∈g,)B1g,ω(ξ=)B,ω(ξg=)B,ωg(ξwisesetmeanshwhicsatisfying)S(9→Ω:ξtelemenrandomaisSonmeasurerandom-stationaryGaandwflotheundertarianvinisCthatassumeeWdefinition:andassumptionwingfollotheatesmotivevoabtionedmensettingcanonicalThe.ertiesopflow-prcalledoftenare(2.26)G,∈h,g,hg=hgand,Ω∈ω,ω=ωewflothisofertiespropstructuralinducedtheandliterature,theinΩonflowastoreferredis}G∈g:g{familyThesetting.thisinariablesvrandomtheorymeasureintselopmendevtrecenandtalsundamenF2:Chapter28ξ(gω)=gξ(ω),g∈G,ω∈Ω,

(gω)=g(ω),ω∈Ω,g∈G,

.3.1.2tionSubsec-inernelskersionvineectivrespanderationsopofexamplesegivthenshalleW3.1].Theorem,31[inconstructionteleganhisoutabmorereadtowishtmighreaderterestedintheandoursfromtdiffereneryvisernelkthisconstructingforhapproacerg’sbKallenell.wastimesametheoutabatactionsgroupenon-transitivossiblypforernelkersionvintheofexistencetheestablishedLast)andtnerGenoftlyenden(indepergbKallen]31[Inassumptions.yregularitologicaltopsomewithspacesologicaltopongroupsologicaltopoferationsopetransitivi.e.spaces,homogeneousofsettingtheinestablished]60[ahleZandRotheryberpapainearedappfirsternelkThisthesis.thisofpartoriginalanconstitutesusthand]21[erpapourfromentakishwhic3.1.1SubsectioninkernelinversiontheofconstructionaegivwilleWRhdRbaRhiVedlId..).5UMfjRh

..).)5edijhkOjVedeSjURVdlRhiVedaRhdRb
IntothehandlefollowingstabilizersTheorem3.1wandeintrotheirduceGcosetsainkκinernelwithinfromtegralsS×StowithGthatrespectwilltoHaarenableusmeasure
λonG.Thiskernelsatisfies
f(gs,g)λ(dg)=f(t,)gκs,t(dg)µs(dt),f∈(S
G)+,s∈S.(3.1)
∫∫∫InparticularκdisintegratestheHaarλonmeasureGalongeachorbitvia
f(g)λ(dg)=f(g)κs,t(dg)µs(dt),f∈G+,s∈S.(3.2)
∫∫∫uniqueTPMadMTkernelκ-.(infr(vomSersion×SktoGernel).IfGopsatisfyinger3.1ates)(prandoperlywithonthetheBorpropelspertiesacSethereisa
(i)κs,gt=κs,tg1,g∈G,s,t∈S,
(ii)κs,tisconcentratedGons,t:={g∈G:gs=t}fort∈Gs,s∈S,
(iii)κs,t(G)=1,t∈Gs,s∈S.

toThisfurtherchapterconstructionsisdevotedandtotheconclusionsconstructionbasedonofthisanimpkernelortan3.2.tinkernel3.1SectionandinSection

IdlRhiVedaRhdRbMdPMffbVOMjVedi

,G∈B,S∈s,)B(,s)s(κ7→)Bs,(maptheβfunctionechoicableasuru-medciateassowithOsystemasuchFixingexists.esentativeseprrorbitofsystemablesura-meathatassumeanderopprebS→,GetLersion).vone-parametric(-.)4adaSSFdy.elyexclusivformwingfollotheinernelkersionvintheusewillewthesisthisInparticularinThenerties.propdesiredthewithκ˜ernelkanotheristhereosesuppκofuniquenessevprooT.Gs∈tfor1=)G(s,tκimpliesthis(i)ybAgainstep.lastthein(i)appliedewwhere(3.3),S∈s,ksµ)G(s,sκ=)dt(sµ)G(s,tκ)t(k∫=ksµyields)3.1(in)t(k:=)gt,(fsettingthatnoteand)2.3(inaskosehoc(iii)orF).G∈hsomeforht=t˜thenGs∈t˜(ifGs∈tallfor0=)s,tcG(s,tκ(i)ybthenButholds.0=)s,tcG(s,tκthathsucGs∈tsomekpicymaewS∈sheacfor0≠sµsinceand,S∈s,S∈t-a.e.sµ,0=)s,tcG(s,tκthatmeansThis.0=)dg(λ}sg≠sg{(∫=)dt(sµ)dg(s,tκ}t≠sg{(∫∫indicators)tanrelevtheofymeasurabilitinsureshwhicSofyertpropBorelthe(and)3.1(ybS∈sforthatnote(ii)orF(iii):and(ii)fulfillsκthatwshotoremainsItfulfilled.are(i)yertpropariancevintheand)3.1(thathsucGtoS×SfromκernelkaobtaintoMernelkthetosµ:=sνandG:=T,S:=Rwith2.17Lemmaapplyymaewmeasurestarianv-inG-finiteσaresµthesinceand)G×·(sM=s1πλ=sµthatclearisiturtherF.G×Sonmeasuretarianv-inGtlyjoinaissMeryevthatyertpropthehasandS→,Gofernessproptheyb-finiteσmeasurablyclearlyishwhicG×StoSfrom,S∈s,)dg(λ·}∈)gs,g({(∫:=sMernelktheConsiderof.oPrapplicationsandernelkersionvIn3:Chapter30

isau-kernelfrSomtoG.
Proof.Justnotethatthemaps,(t)7→κs,t(B)ismeasurableaccordingto3.1Theorem
whiles7→(β(s),s)isu-measurablebyLemma2.8andelementarypropertiesoftheproduct
-algebra.σ

f(t,)gκs,t(dg)µs(dt)=f(t,)gκ˜s,t(dg)µs(dt),f∈(S
G)+,s∈S.
∫∫∫∫SinceGisBorel(inGparticulariscountablygenerated)thisimplies
κs,t=˜κs,t,µs-a.e.t∈S,s∈S,
byand(ii)theinwvemayarianceconcludeκ=prop˜κertysince(i)Gs,tκ=of;andbforκoth˜tallyields̸∈κGss,t.=˜κs,t,t∈Gs,s∈S.Finally,

3.1Inversionkernel31
..).-GfROVMbefRhMjVediMdPhRifROjVlRVdlRhiVedaRhdRbi
Wspeectivwilleininvversionestigatekspernelsecialinthiscasesandsection.examplesofproperoperationsalongwiththeirre-
6kFTbSM-.-(transitiveAcase).firststepofspecializationistoassumethattheopera-
GtionG≠,;→forSissall∈Stransitivthee.correspc:=Hereβ(sonding),s∈S,measureisκ:=justκoneisnevsingleer0.represenIfintativeadditionandsincethe
c,ssc,sstabilizerGislocallycompact(whichisGinheritedforinstancefromGifislocally
close,di.e.c,ctheintersectionofanopenandaclosedG,seesubsetalsoof10[,c,cI.65]),then
κcisnothingbutHaarmeasurewithtotalGmassc,c1onandκsisatranslateofthis
measurerepresentingtheGfromuniformlydistributedmassshiftedontoGthe.coset
Notethatthisimpliesthatc,cnecessarilyapropertransitiveoperationc,swhichistopologically
well-behavinginthesensethatstabilizersareatleastlocallyclosed,musthavecompact
astabilizersdetailedproasoftheseofacarryageneralizationfiniteHaarofmeasure.thisWestatemenrefertintothethefollo3.10generalforwingnon-transitivCorollarye
setting.6kFTbSM-..(groupcase).AssumethelcscGoperatesonitselfvialeft-translation.This
asisforclearlycompactaK⊂Gtransitivtheesetsopπ1(erationK)=Kswhic1hareiseventriviallycontinuous.againItiscompact.clearlyAlsotoptheologically
s1csetshoGosee,sO=={{se}},,sβ∈(g)G,=e,areg∈G,compact.andsinceHereGwe=ha{sµv}se,=s∈(sG,w)eλ,hasv∈eG.Furtherwemay
e,sκs=κe,s=δs,s∈S.
6kFTbSM-.0(trivialoperation).ConsiderthetrivialopG=eration{e},→SwhereS
isanarbitrarymeasurablespace.Thisoperationisclearly{e}ispropercompactsince
andwehaveO=S(thereisonlythiscλ=δhoice)e,β(s)=s,andµs=δs,s∈S.Further
G(s),s=Gs,s={e}=Gandtheinversionkernelreducesto
κ(s),s=κs,s=δe,s∈S.
Itisclearthatanymeasureorkernelisinvariantwithrespecttothisoperation.
coun6kFTbSMtable-.1group(counendowtableedG).withConsiderthethediscretetopmeasurableSologyanopandGarbitraryeration,→SwheremeasurableGisaspace.
HeretheHaarmeasuresareallconstantmultiplesofcountingmeasure(whichisclearly
alsoright-invariant,henceGanyissuchunimodular).Choλosingascountingmeasure
onG,weget∫
µs=({gs∈·}λ(dg)=δgs=jGs,tjδt=jGs,sjδt,s∈S.
∑∑∑g2Gt2Gst2Gs
HereweusedinthePlaststepleft-invarianceofcounting.Aswemayclearlyfindafunction
kjG>s,sj0<on∞,Ss∈sucSh(notethatt2thatGsk<0(tj)G<s,sj∞,s,∈s∈S,S,sincethise∈Gmeanss,s).InthatG,this→Sissettingproperevidenifftly
theinversionkernelisgivenby
1κ(s),s=jG(s),