- #1

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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 by

*F*_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 ∩_αεI

*F*_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 and

*F*_1

for I= 0 and 1,

∩_αεI

*F*_α =

*F*_0 ∩

*F*_1 = {∅, Ω} ∩ {∅, Ω, A_H, A_T} = {∅, Ω} ----(BETA)

the proof says that A_1, A_2,....., A_n,.... ε ∩_αεI

*F*_α , and if we consider A_1 as A_H and A_2 as A_T then why are they not in the intersection of

*F*_0 ∩

*F*_1 as shown in (BETA) ?

Thanks in advance.