By Steven Kalikow

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

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So that the measures of the Si are summable, and with insignificant error sets. Assume the heights of these towers are N1 , N2 , . . Let Pi be a partition of the whole space, consisting of the complement of Si together with the following partition of Si : two points x and y are in the same atom if and only if for all k ∈ {1, 2, . . 6. Birkhoff ergodic theorem and the strong law 39 Ni digits in their binary expansions. The superimposition of the Pi supplies the desired partition; error sets can be obnoxious but Borel–Cantelli can dispose of all but finitely many of them.

2. Constructing measure-preserving transformations In this subchapter we outline three basic ways to construct measure-preserving transformations. For the purposes of this book, only the second and third are essential. 138. First method: explicitly defined functions. Take to be some set, say for example [0, 1), and define some function T : → by an explicit formula, for example, T x = e x mod 1. This T doesn’t preserve Lebesgue measure, but it does preserve some measure, for example the point measure23 on any point y that is a solution to the equation x = e x mod 1.

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.