Billingsley Dimension in Probability Spaces (Lecture Notes by H. Cajar

By H. Cajar

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Then ~1 is again a W-distribution on (y, I) and thus vI : = ~1 E IT. In order to show that Ul is contained in the face E(u) we distinguish two cases: If a = 1, then Ul = u and trivially, ul E E(u). - ~2 on (X, ~) such that V = aVI + (1 - a)U2' Hence we have ul E E(u) also in this case. For each cylinder B E Z and each point ln u (8) Iln In /(B) < E. Z E YI we have 1 Z After solving this inequality for uz (8), integrating with respect to z relative to the H-distribution ~I and reconverting the result one obtains 1n 11 n I n u (B) u~ (B) 1 < E V B E Z.

Here the case "0";; -In 0 +(- lnoo)" has to be considered separately. Hence at most one of the two terms on the right-hand side of the inequality is negative and it does not exceed the other term in absolute value. 1, v) = sUPXEX max {- l~(x), - l~(x)}. 1") V 11, by a simple computation. 1. 1(Z (x)) q(ll, v) = sup lim sup Iln n I XEX nfOO ln v(Zn(x)) lettin~ This identity shows that, given two W-measures with finite q-distance, either both belong to TIna or none of them does. e. they have the same null-cylinders.

B. The metric q* In this part of Section 4 we are goin0 to consider those W-measures y and P which are expressible as integrals over Markov kernels. A. and to new lower bounds for families of W-measures. In the sequel let (Y, 1) be any measurable space and let K be a Markov kernel of (Y, relative to (X, ~). Thus the following propositions are true (see Bauer [3J, § 56): (K1) K is a mapping of Y x -X intolR+; 0 . ) ~). To a large extent, Y may be identified with a subset of IT in view of property (K3).

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