Effectively approximating measurable sets by open sets

We answer a recent question of Bienvenu, Muchnik, Shen, and Vereshchagin. In particular, we prove an effective version of the standard fact from analysis which says that, for any ε > 0 and any Lebesgue-measurable subset of Cantor space, X ⊆ 2, there is an open set Uε ⊆ 2, Uε ⊇ X, such that μ(Uε) ≤ μ(X) + ε, where μ(Z) denotes the Lebesgue measure of Z ⊆ 2. More specifically, our main result shows that for any given rational numbers 0 ≤ ε < ε′ ≤ 1, and uniformly computably enumerable sequence {Un}n∈ω of Σ1-classes such that (∀n)[μ(Un) ≤ ε], there exists a Σ 0,∅′ 1 -class, Y , such that Y ⊇ lim infn Un, and μ(Y ) ≤ ε′. Moreover, Y can be obtained uniformly from ε, ε′, and a u.c.e. index for {Un}n∈ω. We also determine the truth-values of several modifications of our main result, showing that several similar, but stronger, statements are false.