By Peter Barlow

ISBN-10: 1429700467

ISBN-13: 9781429700467

Barlow P. An straight forward research of the idea of numbers (Cornell collage Library, 1811)(ISBN 1429700467)

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**Example text**

Denote the ath coordinate of x E by x, = x,(x). l X,,(W)) E El). 22) Then, (r,23, p) is said to be a sample probability measure space associated with the stochastic process {x,(w), a E %}(w E Q) on 6,and {x,(x), a E 'u} the corresponding sample process. 6. For any stochastic process, there exists an associated sample probability measure space. PROOF. 6. *, XNn(W)) E El). , a,} C 'u} is consistent. Let I' = XirelIR, . 16) holds. 22). ] We conclude this section by proving a lemma, required later, which provides a criterion for the equivalence or singularity of probability measures in terms of the equivalence or singularity of their associated sample probability measures.

If p, is countably additive, then it can be extended to a measure p on the O-ring 23. For the simple case described in the above example, p, is always countably additive. I n the following, we shall investigate the countable additivity of po when the rA are topological spaces. 5, Let A be a directed set, and ( r AZA), , h EA a family of topological spaces. ': r,,-+ r, , A, A' E A, h < A' be a family of continuous mappings satisfying the consistency condition I. 1). Then, (P:'} is said to be a consistent family of projections in the family of topological spaces {(FA , &), A E A}.

24) Since { x a ( - ) , 01 E N} is a joint determining set for 6, , k = 1, 2, and {cp-l(B) I B E S}is the smallest a-algebra in Q with respect to which all the functions {xol(-),c u N] ~ are measurable, it follows that, for any E E 8, there is a B E 23 such that P,((E - v-l(B))u (cp-l(B) - E ) ) = 0, k = 1 , 2. 25) Assume that pl Q pz , P2(E) = 0; we must prove that Pl(E) = 0. 25) holds. 25), p 2 ( B ) = P,(E) = 0. Hence, p l ( B ) = 0. 25), we get P,(E) = p l ( B ) = 0. Therefore, PI Q P, . The remainder of the proof is left to the reader.

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