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Additional info for Billingsley Dimension in Probability Spaces

Example text

Y, 0) = 2 we then 1 see that the lower bound just obtained is indeed the lower limit of the sequence {~ln P(Zn(x))}. Hence we have for the W-measure )J introduced above, )J()_ p x - In 3 + lnZ (l/Z) In 2 V x E B. IZ the relation P-dim(B) = ln 3 +1(l/2) In 2 . Now it can be verified that i(P 1+P Z)-dim(B) > max {P 1-dim(B), PZ-dim(B)}. b). C). 16. ) and suppose there exists a decomposition {Xi with the following property: ViE N v x E Xi 3 n E~ P(Zn(x)) = aiPi(Zn(x)). Then v M c X. P-dim(M) = sup P1,-din(M) iEN I E N} of X Proof.

Now let E E (0,1), E ~ and s > h-dim(M) be fixed. ) which are completely contained in [0,1) and which satis1 1 fy the inequality 5 (3) LiEN (b i - a i ) < t. Now let i E ~ also be an arbitrary but fixed number. Then there exists, by Condition (B) ,for each point x E M~ n (ai' bi ), a smallest number nIx) E IN such that (4) Thus (5) Zn(x)(x) c (ai' bi )· 37 and each interval of the form Zn(x)_l(x) contains at least one of the points a i or bi . Hence there are at least two maximal intervals B~1 and B~1 (where possibly B~1 = B~) of 1 the form In(x)_l(x) for which the following two assertions are valid: (6 ) t: 1 2 Mln(ai,bi)cBiUBi ( 7) A(B ji ) = A(l n(x)-l ( x )) < A(l nIx) (x))l-t: < (b i - a )1-t: i for = 1, 2 and suitable x x(j).

5. ,----T:"":""';"T < n~ ln P(Zn(x)) Then P-dim(l1) Proof. For lim n-><>o 00 v [11] x E X. sup [P-dim(v)] . 2 ~ln l1(Zn(x)) = lim ~ln v(Zn(x)) = - E(v) n-><>o v [vJ x E X, which implies by means of either Condition (1) or (2) that ¥(x) = ~(x) V[v]XEX. 11 The W-distribution ~ of the invariant W-measure 11 is non-atomic if and only if there exists, for every ergodic measure v, a measurable set Xv c X for which v(Xvl = 1 and l1(Xvl = O. 51 The following theorem states in substance that, on the other hand, w-dim(Xv) is large in general.