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By Yasui Y.

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Thus limsupk^oo Qk(Qt)^2/n for each n. This last implies that the family of stationary measures {Qk} is tight on [0,1]. As noted earlier, this means the family {Qk} is tight in Ms((l). Thus, if Q is any limit point, we must have Q e A n A,*, since A H Aoo is closed, and also Q e G since G is closed. Hence, we have contradicted the assumption. , J(A)S= -infQ6AH(Q), but only under the additional hypothesis that the family AM of one-dimensional marginals of Q as Q varies over A forms a tight family of measures in X.

For each f > 0 and each xeX, we use this mapping to induce a probability measure Ttx on Ms(£l) by defining Ftx = P0>XR^, i-e-> if B<^MS(CI), then If P0x is ergodic with invariant measure v(dx) on X, and if Q&Ms(fl) is the stationary Markov process with v as its marginal distribution, then, by the ergodic theorem, as t —» °o, a4 SECTION 9 Here, 8$ is the Dirac measure on Ms(fl) concentrated at Q. We are concerned with the probabilities of large deviations for the measure Ft x. In Section 10 we define, for any Q&Ms(fy, the entropy, H(Q), of the stationary process O with respect to the Markov process P0>x.

E(£l)<= Ms(Cl) be the subset of Ms(£l) consisting of the ergodic measures. ME(fl) is the set of extremals of Ms(fl). It follows from an argument of Oxtoby [11] that there exist a subset ft0c=ft which is ^o°° measurable and a ^o°° measurable map TT^: fto —>ME(ty with the following properties: O(ft0) = 1 for all QeMs(fl) and Q{o>: ir« = Q}= 1 for all QME(fl). d. of Q given &Q°°. d. can always be selected so that it is jointly measurable in Q and w. We define the desired ^ by R^ = R^, o>). Then, for all Qe^ E (ft), SOME PROPERTIES OF ENTROPY 39 But, R^ = R^v, «) is indepenednt of Q, and so we have a linear relation true for extremals and hence true for QeJ(s(ft).

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