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Ergodic Theorems of Birkhoff and von Neumann
Mixing Conditions and Their Characterisations
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A G B a-algebra admits a Borel Bernoulli shift Birkhoff Borel cross-section Borel isomorphism Borel set bounded called complete separable metric compressible Corollary countably additive cr-ideal cr-invariant define denote disjoint union eigenfunction eigenvalue ergodic theorem ergodic with respect exists finite flow built follows function G-action hence homeomorphism incompressible induced automorphism infinitely many negative infinitely many positive intersection isomorphic Kakutani Kechris Lebesgue measure m-null sets measurable sets measure preserving automorphism measure theoretic measure zero metric space metrically isomorphic natural density Neumann automorphism non-empty null sets one-one open set orb(x orbit equivalent pairwise disjoint partition Poincare recurrence lemma Polish topology positive integer probability measure probability space Proof property of Baire proved recurrent point restriction sequence sn(x standard Borel space theory wandering set weak von Neumann weakly mixing write