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1、SYMMETRICMEASURESONCARTESIANPRODUCTS^)BYEDWINHEWITTANDLEONARDJ.SAVAGE1.Introduction.Thispaperhasitsorigininthetheoryofprobability,butwethinkitmaybeofinteresttosomewhoarenotfamiliarwithprob-abilistictechniqueandjargon.Accordingly,weuseforthemostpartthelanguageofmeasuretheo
2、ryinsteadofthelanguageofprobability.However,aninformalprobabilitisticstatementoftheproblemwill,wehope,pavethewayforallreaders.Supposethat,foreachvalueirofaparameter,{en}^°_iisasequenceofrandomvariablesthatarestatisticallyindependentandsubjecttoacommondistributiondepending
3、onir.Ifnowiritselfisarandomvariable,considertheover-alldistributionofthesequence{e?j"_i,i.e.,theaveragewithre-specttoiroftheconditionaldistributionofthesequencegivenir.Thisover-alldistributionwillnotingeneralrendertheen'sindependent,asthecon-ditionaldistributionsgivenware
4、assumedtodo.Nonetheless,itwillobvi-ously,likeadistributionwithindependenten's,beinvariantunderfiniteper-mutationsofthevariablesenamongthemselves,orsymmetric,asweshallsay.Conversely,itistrueunderverygeneralcircumstancesthatanysym-metricdistributiononthee?'scanbeconstructed
5、fromasuitablefamilyofindependentdistributions,parametrizedsaybyir,andasuitabledistributionofir.JulesHaagseemstohavebeenthefirstauthortodiscusssymmetricsequencesofrandomvariables(see[13]).Thispaperdealsonlywith2-valuedrandomvariables.Ithintsat,butdoesnotrigorouslystateorpr
6、ove,therepresentationtheoremforthiscase.Somewhatlater,symmetricdistribu-tionswereindependentlyintroducedbydeFinetti:andtherepresentationtheoremforsymmetricdistributionswasprovedandexploitedbyhim,es-peciallyinconnectionwiththefoundationsofprobability,firstincasetheen'sare2
7、-valuedrandomvariables[10;ll].ThiscasehasalsobeentreatedbyHincin,inmuchthesamemannerasbydeFinetti[18;19].DeFinettialsoprovedthetheoremforthecaseinwhichtheen'sarerealrandomvari-ables[ll].ThiscasehasalsobeentreatedbyDynkin[9],apparentlywith-outknowledgeofdeFinetti'sworkin[l
8、l].Dynkin'stechniquealsoappliestorandomvariablesonallspacesthatare,inacertainsense,separable.Afe
