DOIAbstract
AbstractWe 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 Uε⊆2ω,Uε⊇X, 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 Σ10-classes such that (∀n)[μ(Un)≤ε], there exists a Σ10,0̸′-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 det...
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