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Algorithmic randomness, reverse mathematics, and the dominated convergence theorem

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posted on 01.05.2012, 00:00 by Jeremy AvigadJeremy Avigad, Edward T. Dean, Jason Rute

We analyze the pointwise convergence of a sequence of computable elements ofL1(2ω) 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 conjunction of the statement asserting the existence of a 2-random relative to any given set and the principle of Σ2 collection.

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This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version is available at http://dx.doi.org/10.1016/j.apal.2012.05.010

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01/05/2012

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