2. Caratheodory’s Extension - Probability Tutorials

tutorial 2 : caratheodory s Extension12. caratheodory s ExtensionIn the following, is a set. Whenever a union of sets is denoted asopposed to , it indicates that the sets involved are pairwise 6 Asemi-ringon is a subsetSof the power setP( )with the following properties:(i) S(ii)A, B S A B S(iii)A, B S n 0, Ai S:A\B=n i=1 AiThe last property (iii) says that wheneverA, B S,thereisn 0andA1,..,AninSwhich are pairwise disjoint, such thatA\B=A1 .. 0, it is understood that the corresponding unionis equal to , (in which caseA B). 2: caratheodory s Extension2 Definition 7 Aringon is a subsetRof the power setP( )withthe following properties:(i) R(ii)A, B R A B R(iii)A, B R A\B RExercise thatA B=A\(A\B) and therefore that aring is closed under pairwise that a ring on is also a semi-ring on .Exercise that a set can be decomposed as =A1 A2 A3whereA1,A2andA3are distinct from and . DefineS1 ={ ,A1,A2,A3, }andS2 ={ ,A1,A2 A3, }. Show thatS1andS2are semi-rings on , but thatS1 S2fails to be a semi-ringon.

Tutorial 2: Caratheodory’s Extension 1 2. Caratheodory’s Extension In the following, Ω is a set. Whenever a union of sets is denoted as opposed to ∪, it indicates that the sets involved are pairwise disjoint. Definition 6 A semi-ring on Ω is a subset S of the power set P(Ω) with the following properties:

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