Effectively approximating measurable sets by open sets

We examine 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⊆2ω, there is an open set , such that μ(Uε)≤μ(X)+ε, where μ(Z) denotes the Lebesgue measure of Z⊆2ω, that arises naturally in the context of algorithmic randomness.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 -classes such that (∀n)[μ(Un)≤ε], there exists a -class, Y, such that Y⊇lim infnUn, and μ(Y)≤ε′. Moreover, Y can be obtained uniformly from ε,ε′, and a u.c.e. index for {Un}n∈ω. This answers a recent question of Bienvenu, Muchnik, Shen, and Vereshchagin. We also determine the truth-values of several modifications of our main result, showing that several similar, but stronger, statements are false.