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By J. Pierlo, W. Pfanzagl

Lecture notes in arithmetic No.16

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Example text

In the same man~er you show that for any non-increasing sequence (An)n_ 1 2 ~ ~ - ' '''" the set N An n=l belongs to J~, which concludes the proof. 9) Let J~; be the Borel - algebra of a tight topological Hausdorff space with countable base. = 41 Let P/~* be a p-measure for any sub-G-algebra probability one; on a ~ - a l g e b r a J ~ * ~ ; . ~* o f ~ * Then a regular conditional given ~* exists. 31): be a separable G-algebra on the space Y and let T: X * Y be ~ , ~ * t h a t ~ ~ is compact P ~ * - measurable.

L. N. Grundbegriffe der Wahrscheinlichkeitsrechnung, 1933 +) Quotations are given for the English translations Berlin ~8 Loire, M. Probability theory, l~inceton (New Jersey) 1963 ~arczewski, E. On compact measures, Fundamenta ~athematicae, (1953), pp 113 - 124 Ryll - Nardzewski, C. On quasi - compact measures, Fundamenta Mathematicae, vol. 40 (1953), pp 125 - 130 Sierpinski, vol. @0 W.

And ( 39 such that An, k ~ Cn, k ~ A n and P(A n - An, k ) < 1 for each k ~ 1,2,... 5). Let " u ( & , k)n,k=l , 2 , . . ). Of course, J~2~J~*is countable. /~*). 4) and Lemma (7-7) we can find P/~* , n u l l sets Ni, i m 1,2,3,4 such that (i) for all x$ N 1 0 ~ Po(A,x/g*) ~ 1 for all A ~ 2 (ii) for all x @ N 2 Po(X,x/~*) = 1 (iii) for all x ~ N 3 Po(A'+ A ~ ,x/~*) - Po(f,x/~*) + Po(A ~ ,x/~*) for all disjoint ~,~'m ~2 (iv) for all x~ N 4 Po(An,X/~*) ~ sup Po(An,k,X/~*) k We define for each A m ~ * : I Po(A,x/~*) PI(A'x) : -- P(A) for all A n m ~ 1.

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