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1、Lawoflargenumbersfornon-additivemeasuresYannR′ebill′e?AbstractOuraimistogiveforsomeclassesnon-additivemeasuressomelimittheorems.Forbalancedgamesweobtainaweakandstronglawoflargenumbersforboundedrandomvariables,asharperconclusionisobtainwithexactgames.Weprovideanextensiontoupperenveloppemeasures.Keywo
2、rds:Lawoflargenumbers,exactgames,multi-priormodel.AMSClassi?cation:28C15,91A121IntroductionNon-additivemeasuresarenowadaysstudiedindi?erent?eldofexpertise.Theyareoriginallyknownascapacitiesinpotentialtheory([4]),gameswithtransfer-ableutilitiyincooperativegametheory,fuzzymeasuresinarti?cialintelligen
3、ceorarXiv:0801.0984v1[math.PR]7Jan2008impreciseprobabilitiesinstatistics.SinceKolmogorov’s([8])axiomatictreatiseonprobability,themeasuretheoreticapproachbecamethestandardframework.σ-additivemeasuresturnedouttobetheappropriateobjectstomodelrandomphenomenon.Amajorrequirementforagoodprobabilitytheoryis
4、tobeabletogiveafrequentistjusti?cationtoprobabilitynumbersvialimitfrequenciesorequivalentlylawsoflargenumbersshouldhold.Weaddressthisquestionfornon-additivemeasures.Animportantclassofgameswhichcontainsomeverymildadditivityconditionsarebalancedgames([2],[12])andwithmorestructure,ex-actgames([13]).Thi
5、sgamesareparticularlyimportantsincetheyintroducethecore,akeyconcepttounderstandthegeometryofagame.AnaturalapproachistointroducetheusualMarkov’sconditionstoobtainweakandstronglawoflargenumbersforbalancedandexactgames.Ourresultscanbesharpenedthroughupperintegrals.Thereinterestforupperintegralsrelyonth
6、epossibilitytodeal?Universit′eParisI,CERMSEM,106-112boulevarddel’Hopital,75647ParisCedex13,France.E-mailaddress:yann.rebille@noos.fr12withsetofmeasures.Thisgivesamore?exibletreatementofuncertaintyindeci-sionmakingtheoryasinthemulti-priormodelofGilboa-Schmeidler(1989)([7],see[3]forσ-measures).Ourappr
7、oachismoreelementaryanddeparturesfromtheexistingtopologicalresults,wherepowerfulanalyticalmethodsareused,seeMarinacci([10])forcompactspacesandMaccheroni-Marinacci([9])forpolishspaces.2De?nitionLet(?,A
