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  1.  39
    Algorithmic randomness, reverse mathematics, and the dominated convergence theorem.Jeremy Avigad, Edward T. Dean & Jason Rute - 2012 - Annals of Pure and Applied Logic 163 (12):1854-1864.
    We analyze the pointwise convergence of a sequence of computable elements of L1 in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA0, each is equivalent to the assertion that every Gδ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak Königʼs lemma relativized to the Turing jump of any set. It is also equivalent to the (...)
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  2.  15
    Computable randomness and betting for computable probability spaces.Jason Rute - 2016 - Mathematical Logic Quarterly 62 (4-5):335-366.
    Unlike Martin‐Löf randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and further, provides a general method for abstracting “bit‐wise” definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previously known results. The paper contains a (...)
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  3.  13
    A metastable dominated convergence theorem.Jeremy Avigad, Edward T. Dean & Jason Rute - unknown
    The dominated convergence theorem implies that if is a sequence of functions on a probability space taking values in the interval [0, 1], and converges pointwise a.e., then converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion that is independent of the sequence and the underlying (...)
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  4.  15
    Oscillation and the mean ergodic theorem for uniformly convex Banach spaces.Jeremy Avigad & Jason Rute - unknown
    Let B be a p-uniformly convex Banach space, with p≥2. Let T be a linear operator on B, and let Anx denote the ergodic average ∑i.
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