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Lnj + hn(y, 1) • l~ v x E B. By forming quotients we obtain from the last two equations the relation l' . ;,n,::2---'-r-1~'-'-"--'----rl~~n hn(y,O) ln 3 + hn(y,l) In 6 )J () PI x 1n Z (liS) In 3 + (4/S) In 6 1n 2 1n 3 + ( 4/ 5) 1n 2 for each point x E B. lZ, P1-dim(B) " In 3 + (4/S) In 2 . In the same way it is shown that PZ-dim(B) = In 3 + (~/S) In 2' Now let P := iP1 + ipz. - k(n) x E B and each n E~ we then obtain P(Zn(x)) = n· hn(y, 0). For each point i P1(Zn(x)) + ¥2(Zn(x)) _ 1 (l)k (l)n-k 1 (l)k (_31)n-k -236+26 i ; ;.

3 for, if the dimension system (X, {Z }) is not P-complete then it can * always be embedded in a complete dimension system n(X * , {Zn}). This is accomplished by letting v n EJ1}, by assignin9 to each cylinder B E Zm the cylinder B* : ~ {{ An} E X* I Am ~ B}, and finally by formin9 the decompositions Zn* : ~ {B * I B E Zn} . The embedding ¢: X ~ X* which we have mentioned is then obtained as ¢(x) :~ {Zn(x)} E. X* V X E X. This mapping ¢ is X-X '" -measurable (where X* is, of course, the G-algebra generated by the decompositions Zn).

Therefore P-dim(~) The following theorem deals with a situation in which the W-measures given in terms of integral representations. 4. For i = 1, 2 let (Y i , r i ) be measurable spaces a~d let Ki be Markov kernels of (Y i , r i ) relative to (X, ~). B). Su£pose there exists a W-measure v ETI which is a q*-adherent point of the W-distribution P as well as a q*-adherent point of the W-distribution 'jJ. Then P-dim(~) = 1. Proof. 6). 1. 3 let v k denote the invariant measure constructed by means 2~ k< of a W-distribution k over the Bernoulli measures.

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