Algorithmic randomness, reverse mathematics, and the dominated convergence theorem

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 König'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.

References in corpus (3)

• Randomness and Differentiability
• Lowness notions, measure and domination
• Dynamical systems, simulation, abstract computation

Cited by in corpus (14)

• Metastable convergence theorems
• Bounded variation and the strength of Helly's selection theorem
• Splittings and disjunctions in Reverse Mathematics
• On the Uniform Computational Content of the Baire Category Theorem
• Some nonstandard equivalences in Reverse Mathematics
• On the close interaction between algorithmic randomness and constructive/computable measure theory
• On the logical strengths of partial solutions to mathematical problems
• The strength of compactness in Computability Theory and Nonstandard Analysis
• Coding is hard
• On The Computability of Perfect Subsets of Sets with Positive Measure
• Where Pigeonhole Principles meet König Lemmas
• Pathwise-random trees and models of second-order arithmetic
• Completely determined Borel sets and measurability
• Measure theory and higher order arithmetic