By Steven Kalikow

ISBN-10: 0511676972

ISBN-13: 9780511676970

ISBN-10: 0511679483

ISBN-13: 9780511679483

ISBN-10: 0511681461

ISBN-13: 9780511681462

ISBN-10: 0511801602

ISBN-13: 9780511801600

ISBN-10: 0521194407

ISBN-13: 9780521194402

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

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**Additional resources for An outline of ergodic theory**

**Example text**

Show that: (a) G is dense in R; (b) R is the disjoint union of A and B = A + π; (c) A − A (and hence B − B as well) does not contain any non-trivial interval centered at 0; hence (d) for any interval I , m ∗ (I ∩ A) = m ∗ (I ∩ B) = m(I ) and m ∗ (I ∩ A) = m ∗ (I ∩ B) = 0. Now that we’ve got a non-measurable set, we construct a stationary process that acts locally like coin tossing, in the sense that the joint distribution of any finite collection of its constituent random variables models coin tossing efficaciously, but acts globally very unlike coin tossing in that if you interpreted it as such, one of the “tosses” would be wholly determined by the others.

Theorem. g. d. process composed of real-valued X i . e. Sketch of proof. Denote by (Z , B, ν) the probability space on which the process is defined. Let n > 0 be large and let = { ni : i ∈ Z}, which we view 36 as a countable alphabet. Next, define f : R → by f (x) = nx n . Then let Yi = f ◦ X i . ∞ is an independent stationary process on a countable 211. Exercise. (Yi )i=−∞ alphabet. • Let ( , A, μ, T ) be the measure-preserving system associated with (Yi ). By Theorem 149, ( , A, μ, T ) is ergodic.

Show that ν is non-singular with respect to μ : that is, if μ (A ) = 0 then ν(A ) = 0. Conclude by the Radon–Nikodym theorem that there is some μ -integrable function g such that ν(A ) = A g dμ for every A ∈A. 115. Definition. The function h = g ◦ π, where g is as constructed above, is called the conditional expectation of f given the σ -algebra B. We denote this function by h = E( f |B). If A ∈ A then the conditional probability of A given the σ -algebra B is just the function P(A|B)(x) = E(1 A |B)(x).