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By Steven Kalikow

An creation to ergodic thought for graduate scholars, and an invaluable reference for the pro mathematician.

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5. Conditional expectation 25 Let X, Y, Z or more be measurable maps from a Lebesgue space ( , A, μ) into a measurable space ( , C). The σ -algebra generated by X, Y, Z is the σ -algebra B(X, Y, Z ) generated by {X −1 (C) : C ∈ C} ∪ {Y −1 (C) : C ∈ C} ∪ {Z −1 (C) : C ∈ C}. 123. Definition. Let ( , A, μ) be a Lebesgue space and let f be an integrable function on . 20 We’ll be needing the following exercises later on. ∞ is a sequence of functions whose sum con124. Exercise. e. verges in a dominated way.

Sketch of proof. Let ( , A, μ, T ) be the system in question. 200. Exercise. Show that we may, without loss of generality, assume that ( , A, μ) is [0, 1] with Lebesgue measure. e. (see Definition 154) such that {T i p : i ∈ Z, p ∈ P} separates points mod 0. For each n ∈ N, let Sn be the base of a Rohlin tower of height n 2 having error set at most n12 in measure. For x, y ∈ Sn , write x ∼ y if T k x and T k y agree in the first n digits of their binary expansions, 0 ≤ k ≤ n. It is easy to see that ∼ is an equivalence relation on Sn and that its equivalence classes are measurable.

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.

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