By Steven Kalikow

An advent to ergodic conception for graduate scholars, and an invaluable reference for the pro mathematician.

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Note: this exercise requires some functional analytic background. ) (a) The space M(X ) of probability measures on A can be identified with {λ ∈ C(X )∗ : C(1) = 1, C( f ) ≥ 0 if f ≥ 0}. Namely, μ is identified with λ when f dμ = λ(μ). (This is the Riesz representation theorem; its proof is non-trivial. ) (b) M(X ) is compact in the weak∗ topology. ) (c) If μ ∈ M(X ), let T μ be the measure defined by f dT μ = T f dμ. Now fix any σ ∈ M(X ) and let μ be any weak∗ limit point of the sequence n 1 n i=1 T σ .

E. 126. Exercise. e. e. e. e. Hint: for (b), let f = P(A|B1 ) and B = B2 . Apply part (a) and the previous exercise. 20 That is, E( f |X, Y, Z ) = E f |B(X, Y, Z ) . e. 1. Systems and homomorphisms In this subchapter, we give basic definitions concerning measure-preserving systems and homomorphisms between them. 127. Definition. Let ( , A, μ) be a probability space and assume that T : → is a measure-preserving transformation. We call the quadruple ( , A, μ, T ) a measure-preserving system. If there are sets X, X ∈ A of full measure such that T is a bimeasurable bijection between X and X then we say that the system ( , A, μ, T ) is invertible, or simply that T is invertible.

Exercise. Show that p extends uniquely to a premeasure on C. Let A and μ be the resulting σ -algebra and measure extending p obtained via Carathéodory’s theorem. Note that ( , A, μ) is a Lebesgue space. 107. Aside. According to the previous theorem, if we’re dealing with a prepro∞ and whatever we would like to know about it is characterized cess (X i )i=−∞ by the joint distributions of finite collections of the X i , we can assume it is a process and hence that the space we are working on is Lebesgue.