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Respected Members,

Suppose Ω is the set of eight possible outcomes of three coin tosses i.e. Ω={{HHH, HHT, HTH, HTTT, THH, THT, TTH, TTT}

So if we are not told the results then the sigma algebra ( denoted byF_α) at position α=0 is

F_0 = {∅, Ω}

Now if are told the first coin toss only then,

A_H={HHH, HHT, HTH, HTTT}, and A_T={THH, THT, TTH, TTT}

which the sigma Algebra at α=1 is

F_1={∅, Ω, A_H, A_T}

now in the attached picture the proof says that

A_1, A_2,....., A_n,.... ε ∩_αεIF_α

if we just consider two sigma algebras for our convenience to check this lets take the intersection of two above coin toss's sigma algebras i.e.F_0 andF_1

for I= 0 and 1,

∩_αεIF_α =F_0 ∩F_1 = {∅, Ω} ∩ {∅, Ω, A_H, A_T} = {∅, Ω} ----(BETA)

the proof says that A_1, A_2,....., A_n,.... ε ∩_αεIF_α , and if we consider A_1 as A_H and A_2 as A_T then why are they not in the intersection ofF_0 ∩F_1 as shown in (BETA) ?

Thanks in advance.

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# Sigma Algebra on Omega (Sample Space)

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