By Steven Kalikow

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

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Definition. A stationary process (X i )i=−∞ is called an independent process if the random variables X i are independent of each other, so that for example P(X 0 = a, X 1 = b, X 2 = c) = P(X 0 = a)P(X 0 = b)P(X 0 = c). 149. Theorem. Any independent process is ergodic. Sketch of Proof. Let ( , A, μ, T ) be the system arising from the process. Suppose that this system is not ergodic. Then there exists A ∈ A with 0 < μ(A) < 1 such that μ(A T −1 A) = 0. By Corollary 71, the algebra of cylinder sets generates A mod 0.

Theorem. Let ( , A, μ) be a probability space and suppose that (Si )i=1 ∞ is a sequence of measurable sets with i=1 μ(Si ) < ∞. Suppose that for each i ∈ N, Pi is a finite measurable partition of having the property that Sic ∈ Pi . ∞ is countable mod 0. Then the superimposition P of (Pi )i=1 Sketch of proof. 194. Exercise. Show that it is sufficient to show that for each finite subfamily P of P such that μ( p∈P p) > 1 − . Let > 0 and choose j such that μ( ∞ i= j > 0, there is a • Si ) < . 195. Exercise.

D) The identity map x → x is a measure-preserving injection from ([0, 1], A, μ) to ([0, 1], L, m) that has a non-measurable inverse. Conclude that ([0, 1], A, μ) is not a Lebesgue space. e. e. x, ∞ ∞ such that x = −i there is a unique {0, 1}-valued sequence (ai )i=1 i=1 ai 2 . ∞ Define (for these values x) a sequence of random variables (Yi )i=−∞ from ([0, 1], A, μ) into the alphabet = {0, 1} as follows: (i) Yi (x) = a2i−1 for i > 0; (ii) Yi (x) = a−2i for i < 0; (iii) Y0 (x) = 1 if x ∈ A and 0 otherwise.

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