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5. 1]. We leave it to the reader as an exercise. 1). Summing it up over A ∈ A for a finite partition A, obtain 0 ≤ Hρ (A) − a Hµ (A) − (1 − a) Hν (A) ≤ log 2. 6(a). 1). Remark. 16. 11. 5 Shannon-Mcmillan-Breiman theorem Let (X, F , µ) be a probability space, let T : X → X be a measure preserving endomorphism of X and let A be a countable finite entropy partition of X. 1 (maximal inequality). For each n = 1, 2, . . let fn = I(A|An1 ) and f ∗ = supn≥1 fn . Then for each λ ∈ R and each A ∈ A µ({x ∈ A : f ∗ (x) > λ}) ≤ e−λ .

MEASURE PRESERVING ENDOMORPHISMS is again in M (F ). The subspace M (F , T ) of M (F ) consisting of T -invariant measures is also convex. Recall that a point in a convex set is said to be extreme if and only if it cannot be represented as a convex combination of two distinct points with corresponding coefficient 0 < α < 1. We shall prove the following. 8. The ergodic measures in M (F , T ) are exactly the extreme points of M (F , T ). Proof. Suppose that µ, µ1 , µ2 ∈ M (F , T ), µ1 = µ2 and µ = αµ1 + (1 − α)µ2 with 0 < α < 1.

11. (a) There exists a finest measurable partition A (mod 0) into T -invariant sets (called the ergodic decomposition). Almost all of its components are ergodic. (b) h(T ) = X/A h(TA ) dµA (A). Proof. The part (a) will not be proved. 5). 58 CHAPTER 1. MEASURE PRESERVING ENDOMORPHISMS To prove the part (b) notice that for every T -invariant measurable partition A, for every finite partition ξ and almost every A ∈ A, writing ξA for the partition {s ∩ A : s ∈ ξ}, we obtain − h(TA , ξA ) = H(ξA |ξA )= A − ) dµA .

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Algebre lineare et geometrie elementaire by Dieudonne J.

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