Algorithmic randomness, reverse mathematics, and the dominated convergence theorem

Details Algorithmic randomness, reverse mathematics, and the dominated convergence theorem Algorithmic randomness, reverse mathematics, and the dominated convergence theorem

2011-06-03

arxiv.org/abs/1106.0775v2

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We analyze the pointwise convergence of a sequence of computable elements of L^1(2^omega) in terms of algorithmic randomness. We then show that, over the base theory RCA_0, a version of the dominated convergence theorem is equivalent to the assertion that every G_delta subset of Cantor space with positive measure has an element. It is also equivalent to a version of weak weak K\"onig's lemma relativized to the Turing jump of any set. These principles imply the existence of a 2-random relative to any given set, and are equivalent to that assertion in the presence of Sigma^0_2 induction.

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