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From the previous lemma we see that ∞ P (F ∆F˜ ) ≤ P (F ∆ ∞ Fi ) + P ( i=1 Fi ∆F˜ ) ≤ . 6. ISOMORPHISM 23 Exercises 1. Show that if m and p are two probability measures on (Ω, σ(F)), where F is a field, then given an arbitrary event F and > 0 there is a field event F0 ∈ F such that m(F ∆F0 ) ≤ and p(F ∆F0 ) ≤ . 6 Isomorphism We have defined random variables or random processes to be equivalent if they have the same output probability spaces, that is, if they have the same probability measure or distribution on their output value measurable space.

Continue in this way filling in the sequence of Gn with enough repeats to meet the condition. The sequence Hn ∈ Fn is thus a nonempty nonincreasing sequence of field elements. Clearly by construction ∞ ∞ Hn = n=1 Gn n=1 and hence we will be done if we can prove this intersection nonempty. We accomplish this by assuming that the intersection is empty and show that this leads to a contradiction with the assumption of a standard space. Roughly speaking, a decreasing set of nonempty field elements collapsing to the empty set must contain a decreasing sequence of nonempty basis atoms collapsing to the empty set and that violates the properties of a basis.

3. 1 Let Fi , i ∈ I, be a family of standard fields for some countable index set I. Let F be the product field generated by all rectangles of the form F = {xI : xi ∈ Fi , i ∈ M}, where Fi ∈ Fi all i and M is any finite subset of I. That is, F = F(rect(Fi , i ∈ I)), then F is also standard. Proof: Since I is countable, we may assume that I = {1, 2, . }. For each i ∈ I, Fi is standard and hence possesses a basis, say {Fi (n), n = 1, 2, . }. Consider the sequence Gn = F(rect(Fi (n), i = 1, 2, . .

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Stoshastic processes and stochastic integration by Pivato M.

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