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arbixerwoxottdrp1l

arbixerwoxottdrp1l

Answered question

2022-04-10

Let ( X n ) n N be a sequence of independent real valued random variables on the probability space ( Ω , F , P ). I'm trying to show that the collection C of events { ω : X i ( ω ) B i , i = 1 , , k }, for B i B ( R ), generates the sigma-algebra H k := σ ( X 1 , , X k ). I've already shown that C is a π-system hence the next step of my proof would be to show that any X i , i = 1 , , k, is σ ( C ) / B ( R ) measurable. But given that any element A C has the form A = { ω : X i ( ω ) B i , i = 1 , , k } = i = 1 k { ω : X i ( ω ) B i } = i = 1 k X i 1 [ B i ], I don't really know how to proceed. We certainly know that for any fixed j, A = i = 1 k X i 1 [ B i ] X j 1 [ B j ], but I can't really convince myself of the fact that then X j 1 [ B j ] σ ( C ). How should I proceed with this proof?

Answer & Explanation

Gallichi5mtwt

Gallichi5mtwt

Beginner2022-04-11Added 18 answers

If B i = R for all i j and B j = B then { ω : X i ( ω ) B i , i = 1 , 2... , k } = X j 1 ( B ) so X j 1 ( B ) C σ ( C ).

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