The Coin Toss

Abstract

Let S = {H,T} be a two element set with members H and T . We will operate on the space of outcomes Ω = SN. This is an indexed set with with ωn ∈ S for each ω ∈ Ω, n ∈ N. The idea is that the nth index of an outcome ω models a bit of information at time n: for example the result of a coin flip. Let Ωn = S n and Fn = ⊗k=1P(S) = P(Ωn). Note that Fn is a σ-algebra. Furthermore, there is a canonical injection of Fn into P(Ω). Define the projection operator Π : Ω → Ωn where Π(ω) is the unique ωn ∈ Ωn such that ωk = ωn k , 1 ≤ k ≤ n. Intuitively, Π “forgets” what happens after time n. The injection from Fn into P(Ω) is then defined by the pre-image of Π. We will identify Fn with Π(Fn), giving us Fn ⊂ P(Ω).