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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:
2、(i)2F(ii)A2F)Ac=4nA2F+[1(iii)An2F;n1)An2Fn=1Tutorial1:Dynkinsystems2Exercise1.LetFbea-algebraon.Showthat;2F,thatifA;B2FthenA[B2FandalsoAB2F.RecallthatBnA=BAcandconcludethatFisalsoadynkinsystemon.Exercise2.Let(Di)i2Ibeanarbitraryfamilyofdynkinsystems4on,
3、withI6=;.ShowthatD=i2IDiisalsoadynkinsystemon.Exercise3.Let(Fi)i2Ibeanarbitraryfamilyof-algebrason,4withI6=;.ShowthatF=i2IFiisalsoa-algebraon.Exercise4.LetAbeasubsetofthepowersetP().De ne:4D(A)=fDdynkinsystemon:ADgShowthatP()isadynkinsystemon,andthatD(
4、A)isnotempty.De ne:4D(A)=DD2D(A)Tutorial1:Dynkinsystems3ShowthatD(A)isadynkinsystemonsuchthatAD(A),andthatitisthesmallestdynkinsystemonwithsuchproperty,(i.e.ifDisadynkinsystemonwithAD,thenD(A)D).De nition3LetAP().WecalldynkinsystemgeneratedbyA,thedynki
5、nsystemon,denotedD(A),equaltotheintersectionofalldynkinsystemson,whichcontainA.Exercise5.Doexactlyasbefore,butreplacingdynkinsystemsby-algebras.De nition4LetAP().Wecall-algebrageneratedbyA,the-algebraon,denoted(A),equaltotheintersectionofall-algebraso
6、n,whichcontainA.De nition5AsubsetAofthepowersetP()iscalleda-systemon,ifandonlyifitisclosedunder niteintersection,i.e.ifithastheproperty:A;B2A)AB2ATutorial1:Dynkinsystems4Exercise6.LetAbea-systemon.ForallA2D(A),wede ne:4 (A)=fB2D(A):AB2D(A)g1.IfA2A,showt
7、hatA (A)2.ShowthatforallA2D(A), (A)isadynkinsystemon.3.ShowthatifA2A,thenD(A) (A).4.ShowthatifB2D(A),thenA (B).5.ShowthatforallB2D(A),D(A) (B).6.ConcludethatD(A)isalsoa-systemon.Exercise7.LetDbeadynkinsystemonwhichisalsoa-system.1.ShowthatifA;B2DthenA
8、[B2D.Tutorial1:Dynkinsystems54n2.LetAn2D;n1.ConsiderBn=[i=1Ai.Showthat+1+1[n=1An=[n=1Bn.3.ShowthatDisa-algebraon.Exercise8.LetAbea-systemon.ExplainwhyD(A)isa-algebraon,and(A)isadynkinsystemon.Conc
