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Example text

MEASURE PRESERVING ENDOMORPHISMS Proof 2. 5). 4, for every A ∈ A, the sequence E(11A |F(Bn )) converges to E(11A |F(B)) in L2 . Hence, for every A ∈ A, the sequence µBn (x) (A∩ Bn (x)) converges to µB( x) (A∩B(x)) in measure µ. 3, this implies the convergence k(µBn (x) (A ∩ Bn (x))) → k(µB(x) (A ∩ B(x))) in measure µ (we do not assume x ∈ A here). Summing over all A ∈ A we obtain the convergence HµBn (x) (A|Bn (x)) → HµB (x) (A|B(x)) in measure µ. 5). e. 5 as follows hµ (T, A) = H(A|A− ), where A− := ∞ T −n (A).

5. In the case of descending sequence Bn the proof is straightforward. In the case of ascending Bn use H(A|Bn )−H(Aj |Bn ) = H(A|(Aj ∨Bn )) ≤ H(A|(Aj ∨B1 )) = H(A|B1 )−H(Aj |B1 ). This implies that the convergence as j → ∞ is uniform with respect to n, hence in the limit H(A|Bn ) → H(A|B). ♣ 56 CHAPTER 1. MEASURE PRESERVING ENDOMORPHISMS Proof 2. 5). 4, for every A ∈ A, the sequence E(11A |F(Bn )) converges to E(11A |F(B)) in L2 . Hence, for every A ∈ A, the sequence µBn (x) (A∩ Bn (x)) converges to µB( x) (A∩B(x)) in measure µ.

And in L . Proof. 4. e. of each fnA , hence of fn . 2. and Dominated Convergence Theorem. 4 (Shannon-McMillan-Breiman). Suppose that A is a countable partition of finite entropy. e. e. and in L1 . 2) Furthermore h(T, A) = lim n→∞ 1 H(An ) = n+1 fI dµ = f dµ. 6), in the context of Lebesgue spaces, where the notion of information function I will be generalized. Proof. First note that the sequence fn = I(A|An1 ), n = 1, 2, ... 3. 5. e. convergence of time averages to fI holds by Birkhoff’s Ergodic Theorem.