Turning Borel sets into clopen sets effectively

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 sho...

A Course on Borel Sets

Decomposing the real line into Borel sets closed under addition

We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in ZFC and even in the theory ZFC + c = ω2 if the number of pieces can be uncountable but less than the continuum. We also ...

Effectively Simple Sets

We use the word "number" to mean positive integer, and "set" to mean set of positive integers. A set a is called simple (after Post [l ]) iff a is r.e. (recursively enumerable), infinite, and its complement à, though infinite, contains no infinite recursively enumerable subset. We shall call a set a effectively simple if a and its complement are infinite, a is r.e., and there exists a recursive...

Borel Sets with Large Squares

For a cardinal μ we give a sufficient condition ⊕μ (involving ranks measuring existence of independent sets) for: ⊗μ: if a Borel set B ⊆ R × R contains a μ-square (i.e. a set of the form A × A, |A| = μ) then it contains a 20 -square and even a perfect square. And also for ⊗′μ: if ψ ∈ Lω1,ω has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute...