An Outline of Ergodic Theory
This informal introduction provides a fresh perspective on isomorphism theory, which is the branch of ergodic theory that explores the conditions under which two measure preserving systems are essentially equivalent. It contains a primer in basic measure theory, proofs of fundamental ergodic theorems, and material on entropy, martingales, Bernoulli processes, and various varieties of mixing. Original proofs of classic theorems - including the Shannon–McMillan–Breiman theorem, the Krieger finite generator theorem, and the Ornstein isomorphism theorem - are presented by degrees, together with helpful hints that encourage the reader to develop the proofs on their own. Hundreds of exercises and open problems are also included, making this an ideal text for graduate courses. Professionals needing a quick review, or seeking a different perspective on the subject, will also value this book.
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algebra approximation assume Bernoulli Bernoulli processes Birkhoff ergodic theorem cells choose close columns conditioned construct convergence coordinates copy Corollary countable alphabet coupling cylinder set deﬁne Deﬁnition denote disjoint distribution entropy equivalence classes ergodic theory error set example exists exponentially fat fat submeasures ﬁnite finite alphabet ﬁrst foregoing exercise function Hamming distance hence Hint Idea of proof independent concatenation independent process induced interval isomorphism theorem Lebesgue measure Lebesgue space lemma Let Q limsup martingale mean Hamming distance measurable partition measurable set measure space measure-preserving system measure-preserving transformation n-distribution notation ordered partition Ornstein P-name points previous exercise probability space random variables reader Rohlin tower rungs sequence Show Sketch of proof stationary process subsequential limit subsets sufﬁciently Suppose T-invariant tower of height union variation distance words of length zero