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Measure Preserving Transformations
Proof of the Proposition
2 other sections not shown
Birkhoff Ergodic Theorem Birkhoff Theorem Bishop Bishop's Upcrossing Inequality bounded non-negative integrable Cauchy sequence characteristic function condition 1.1 constructively true converges almost everywhere Define the set disjoint partitions Doctor of Philosophy Doob Doob's Upcrossing Inequality exceptional values finite subsets fj(x fk(x fn(x fQ(x full subset function and let Improvement on Doob's integer h integrable f integrable sets integrable subsets John Arthur Nuber left translation Lemma Let f Let g linear operator Mathematics maximum integer measurable function measurable set measure preserving transformation measure space n(Ac n(Ak negative integer non-negative integer non-negative integrable function null set pair of non-negative pair of real positive integer preserving transformation satisfying Proof proposition real numbers San Diego satis satisfied classically satisfies 1.3 sequence of averages Submartingale Sequences thesis uniformly for non-negative uniformly integrable unit circle University of California