1. Dynkin systems - Probability

1 Tutorial 1: Dynkin systems11. Dynkin systemsDefinition 1 ADynkin systemon a set is a subsetDof thepower setP( ), with the following properties:(i) D(ii)A, B D,A B B\A D(iii)An D,An An+1,n 1 + n=1An DDefinition 2A -algebraon a set is a subsetFof the powersetP( )with the following properties:(i) F(ii)A F Ac = \A F(iii)An F,n 1 + n=1An 1: Dynkin systems2 Exercise a -algebra on . Show that F,thatifA, B FthenA B Fand alsoA B F. Recall thatB\A=B Acand conclude thatFis also a Dynkin system on .Exercise (Di)i Ibe an arbitrary family of Dynkin systemson , withI = . Show thatD = i IDiis also a Dynkin system on.

2 Exercise (Fi)i Ibe an arbitrary family of -algebras on ,withI = . Show thatF = i IFiis also a -algebra on .Exercise a subset of the power setP( ). Define:D(A) ={DDynkin system on :A D}Show thatP( ) is a Dynkin system on , and thatD(A) :D(A) = D D(A) 1: Dynkin systems3 Show thatD(A) is a Dynkin system on such thatA D(A), andthat it is the smallest Dynkin system on with such property, ( a Dynkin system on withA D,thenD(A) D).Definition 3 LetA P( ).WecallDynkin system generatedbyA, the Dynkin system on ,denotedD(A), equal to the intersectionof all Dynkin systems on , which exactly as before, but replacing Dynkin systems by 4 LetA P( ).

3 Wecall -algebra generatedbyA,the -algebra on ,denoted (A), equal to the intersection of all -algebras on , which 5 AsubsetAof the power setP( )is cal led a -systemon , if and only if it is closed under finite intersection, if it hasthe property:A, B A A B 1: Dynkin systems4 Exercise a - system on . For allA D(A), we define: (A) ={B D(A):A B D(A)}1. IfA A, show thatA (A)2. Show that for allA D(A), (A) is a Dynkin system on .3. Show that ifA A,thenD(A) (A).4. Show that ifB D(A), thenA (B).5. Show that for allB D(A),D(A) (B).6. Conclude thatD(A)isalsoa - system on.

4 Exercise a Dynkin system on which is also a Show that ifA, B DthenA B 1: Dynkin systems52. LetAn D,n 1. ConsiderBn = ni=1Ai. Show that + n=1An= + n= Show thatDis a -algebra on .Exercise a - system on . Explain whyD(A)isa -algebra on , and (A) is a Dynkin system on . Conclude thatD(A)= (A). Prove the theorem:Theorem 1 ( Dynkin system )LetCbe a collection of subsets of which is closed under pairwise intersection. IfDis a Dynkin systemcontainingCthenDalso contains the -algebra (C)generated