Measurable functions and covering properties

Abstract

Let X be a set and A⊆P(X) be a family closed under finite intersections such that ∅,X∈A. If ψ=o,ω,γ, then Ψ(A) is the family of those ψ-covers U for which U⊆A. In [3], properties (Ψ0) of a family F⊆XR of real functions have been introduced. The main result of the paper Theorem 4.1 reads as follows: if Φ=Ω,Γ and Ψ=O,Ω,Γ, then for any pair 〈Φ,Ψ〉 different from 〈Ω,O〉, X has the covering property S1(Φ(A),Ψ(A)) if and only if the family of non-negative upper A-semimeasurable real functions satisfies the selection principle S1(Φ0,Ψ0). Similarly for Sfin and Ufin. Some related results are also presented.