On a conjecture of Dobrinen and Simpson concerning almost everywhere domination

Dobrinen and Simpson [4] introduced the notions of almost everywhere domination and uniform almost everywhere domination to study recursion theoretic analogues of results in set theory concerning domination in generic extensions of transitive models of ZFC and to study regularity properties of the Lebesgue measure on 2 in reverse mathematics. In this article, we examine one of their conjectures concerning these notions. Throughout this article, ≤T denotes Turing reducibility and μ denotes the Lebesgue (or “fair coin”) probability measure on 2 given by μ({X ∈ 2 |X(n) = i}) = 1/2. A property holds almost everywhere or for almost all X ∈ 2 if it holds on a set of measure 1. For f, g ∈ ω, f dominates g if ∃m∀n > m(f(n) > g(n)). Definition 1.1 (Dobrinen, Simpson). A set A ∈ 2 is almost everywhere (a.e.) dominating if for almost all X ∈ 2 and all functions g ≤T X, there is a function f ≤T A such that f dominates g. A is uniformly almost everywhere (u.a.e.) dominating if there is a function f ≤T A such that for almost all X ∈ 2 and all functions g ≤T X, f dominates g.