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1、Tutorial1:Dynkinsystems11.DynkinsystemsDe nition1AdynkinsystemonasetisasubsetDofthepowersetP(),withthefollowingproperties:(i)2D(ii)A;B2D;AB)BnA2D+[1(iii)An2D;AnAn+1;n1)An2Dn=1De nition2A-algebraonasetisasubsetFofthepowersetP()withthefollowingproperties:(i)2F(ii)A2F)Ac=4nA2F+[1
2、(iii)An2F;n1)An2Fn=1Tutorial1:Dynkinsystems2Exercise1.LetFbea-algebraon.Showthat;2F,thatifA;B2FthenA[B2FandalsoAB2F.RecallthatBnA=BAcandconcludethatFisalsoadynkinsystemon.Exercise2.Let(Di)i2Ibeanarbitraryfamilyofdynkinsystems4on,withI6=;.ShowthatD=i2IDiisalsoadynkinsystemon.E
3、xercise3.Let(Fi)i2Ibeanarbitraryfamilyof-algebrason,4withI6=;.ShowthatF=i2IFiisalsoa-algebraon.Exercise4.LetAbeasubsetofthepowersetP().De ne:4D(A)=fDdynkinsystemon:ADgShowthatP()isadynkinsystemon,andthatD(A)isnotempty.De ne:4D(A)=DD2D(A)Tutorial1:Dynkinsystems3ShowthatD(A)isa
4、dynkinsystemonsuchthatAD(A),andthatitisthesmallestdynkinsystemonwithsuchproperty,(i.e.ifDisadynkinsystemonwithAD,thenD(A)D).De nition3LetAP().WecalldynkinsystemgeneratedbyA,thedynkinsystemon,denotedD(A),equaltotheintersectionofalldynkinsystemson,whichcontainA.Exercise5.Doexact
5、lyasbefore,butreplacingdynkinsystemsby-algebras.De nition4LetAP().Wecall-algebrageneratedbyA,the-algebraon,denoted(A),equaltotheintersectionofall-algebrason,whichcontainA.De nition5AsubsetAofthepowersetP()iscalleda-systemon,ifandonlyifitisclosedunder niteintersection,i.e.if
6、ithastheproperty:A;B2A)AB2ATutorial1:Dynkinsystems4Exercise6.LetAbea-systemon.ForallA2D(A),wede ne:4
