Let A 1 , A 2 , … be a collection of events, and let...

mistergoneo7

mistergoneo7

Answered

2022-07-05

Let A 1 , A 2 , be a collection of events, and let A be the smallest σ -field of subsets of Ω which contains all of them. If A A , then there exists a sequence of events { C n } such that
C n A n  and  P ( A C n ) 0  as  n
where A n is the smallest σ -field which contains the finite collection A 1 , A 2 , , A n .
However, I do not know how to choose the C n such that they turn up in σ ( A 1 , . . A n ) ?

Answer & Explanation

Elias Flores

Elias Flores

Expert

2022-07-06Added 24 answers

Union of A n 's is an algebra which generates A . Just apply Halmos' Theorem now. You get C n A k n for some k n incerasing to . But you can keep repeating each C n to make sure that C n A n for each n: Look ar , , , . . . , C 1 , C 1 , . . . C 1 , C 2 , C 2 , . . . , C 2 , . . . where is repeated k 1 1 times, C 1 is repeated k 2 k 1 1 times, and so on.

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