Algorithmic randomness, reverse mathematics, and the dominated convergence theorem

TL;DR

This paper explores the relationship between algorithmic randomness, reverse mathematics, and the dominated convergence theorem, establishing equivalences between measure-theoretic principles and logical systems within RCA_0.

Contribution

It demonstrates that certain forms of the dominated convergence theorem are equivalent to key principles in reverse mathematics and algorithmic randomness over RCA_0.

Findings

Equivalence of dominated convergence formulations to measure-theoretic principles.

Connection between G_delta sets with positive measure and algorithmic randomness.

Identification of logical strength of convergence theorems in reverse mathematics.

Abstract

We analyze the pointwise convergence of a sequence of computable elements of L^1(2^omega) in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA_0, each is equivalent to the assertion that every G_delta subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak K\"onig's 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 Sigma_2 collection.