By Steven Kalikow

ISBN-10: 0521194407

ISBN-13: 9780521194402

This casual advent makes a speciality of the department of ergodic idea referred to as isomorphism conception. routines, open difficulties, and worthy tricks actively have interaction the reader and inspire them to take part in constructing proofs independently. excellent for graduate classes, this booklet can be a precious reference for the pro mathematician.

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Let S1 ∈ S and for n > 1, choose Sn with Sn−1 Sn and μ(Sn ) ≥ [μ(Sn−1 ) + f (Sn−1 )]/2. 31 That is, μ({x : ∃i ∈ N with T i x = x}) = 0. 9780521194402c02 34 CUP/KKW October 10, 2009 21:55 Page-34 Measure-preserving systems 172. Theorem. ) Let ( , A, μ, T ) be a non-periodic measure-preserving system, let N ∈ N and let > 0. There exists some S ∈ A such that S, T S, T 2 S, . . , T N −1 S are pairwise N −1 i T S) < . disjoint and μ(X \ i=0 Idea of proof. Since the space is Lebesgue, we can assume it is the unit interval, endowed with its usual metric.

The map f → E( f |B) on L 2 ( , A, μ) is the orthogonal projection onto the subspace of B measurable, square integrable functions. Sketch of proof. 243. Exercise. Fix f ∈ L 2 ( , A, μ). It suffices to show that for an arbitrary B-measurable, square integrable function g, || f − g|| ≥ || f − E( f |B)||. • Fix such a g. We need one more thing: 244. Exercise. Let (X, C, ν) be an arbitrary probability space and f ∈ L 2 (X ). • Show that infc∈R ||F − c|| = || f − f dν||. Now we have || f − g|| = | f − g|2 dμ = | f − g|2 dμx dμ(x) 2 f − ≤ f dμx dμx dμ(x) 2 = = f − f dμx dμ(x) | f − E( f |B)|2 dμ(x) = || f − E( f |B)||.

Conditional expectation Page-25 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. 21 Show E( i=1 125.

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