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 sho...

This article is a fundamental study in computable measure theory. We use the framework of TTE, the representation approach, where computability on an abstract set X is defined by representing its elements with concrete “names”, possibly countably infinite, over some alphabet Σ. As a basic computability structure we consider a computable measure on a computable σ-algebra. We introduce and compar...

In the context of arbitrary Polish groups, we investigate the Galvin– Mycielski–Solovay characterization of strong measure zero sets as those sets for which a meager collection of right translates cannot cover the