By Yuichiro Kakihara

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And the shift S on X given by S : (... ) >->■(... , X O J S I , . . ) , xjj. = Xfc+i, 4 e Z . ) :xk = x°k, i < k < j} and let Extend /^o to the a-algebra X generated by all cylinder sets, denoted by p.. t. \i and hence (X, X, p, 5) is a dynamical system. The shift S is called a ( p i , . . ,pt)-Bernoulli shift. Since 21 = {[xo = «i], • • • , [^o = ai]} oo is a finite partition of X and Sn2l = X by definition, we have by Theorem 10 V n=—oo and Lemma 5 (2) that H(s) = Hm,s) = hm iff(Vs-fcsa).

Proo/. Let t € M+(X). For any a 6 P(£) and m, n > 1 one has that 4 E ^)loe^) 2n + k < -- 1 £ tiA)]ogtiA) In + fc iie%)va(m) ! , + ,. 5. Entropy functionals = I(k, m, n) + J(m, n), say. 8) For any e > 0 we can choose mo > 1 such that 0 < J{m,n) < J(m,0) < e, m > mo, n > 1. (-A)logf(A) < J(fc,m,n) + e, Aeavat_! If we let k ->■ co, then I(k,m,n) # ( £ ^ , 5 ) . 8) -)• -ff($,a,5)

Let A = {B 6 X : A~B}, the equivalence class containing X e l , and 93^ = {A : A € X}, the set of all equivalence classes of X, called the measure algebra of n. The measure fj, on 93M is of course defined by H(A) = fi(A), A eX and is strictly positive in the sense that n(A) > 0 if A ^ 0. For now we consider conjugacy between a pair of probability measures. Definition 1. Let (Xj, Xj, fij) (j = 1,2) be two probability measure spaces with the associated measure algebras 931 and 2*2, respectively. , T satisfies that H2(TB) = T y OO v OO rs TBc={TBf, M x (B), T / OO \ B e ®i, OO ( u^) = u i' ( n i) = n r i ^ ^ e ® i > J ^ L B Conjugacy between a pair of probability measures can be phrased in terms of a unitary operator between L 2 -spaces as follows.