Outer Measures on the Real Line by Weak Selections

𝜎-ideals and outer measures on the real line

“…Let X ⊆ R and let f be a weak selection. If |R\X| ≤ ω, by Corollary 2.7 from [1], then we obtain that N f ⊕ P(X) = P(R), and Example 3.6 from [1] provides a weak selection g such that N g = P(R). Hence, we will suppose that |R\X| > ω.…”

“…The following easy lemma has been frequently used to construct examples and counterexamples (see for instance [1]). Lemma 1.1.…”

“…According to Lemma 4.7, M f φ = P(R). In particular, if f is the weak selection defined in [1,Ex. 3.6], then λ f (r, s] f = λ fc (r, s] f = 0, for every r, s ∈ R, but λ f = λ fc (this answers negatively Question 3.14 of [1]).…”

“…One important property of the weak selections is that they give the possibility to generates topologies which have interesting topological properties (see [3], [6] and [9]). In the article [1], the authors introduced the notion of f -outer measure by using a weak selection f on the real line as follows: Certainly, the Lebesgue outer measure λ coincides with the outer measure λ f E (briefly denoted by λ) where f E is the weak selection induced by the Euclidean order of R. Given a weak selection f on R, N f will denote the σ-ideal consisting of all λ f -null sets and the family of λ f -measurable subsets will be denoted by M f . In particular, M will stand for the family of Lebesgue measurable sets, and N for the Lebesgue null subsets of R. Since the null sets of an outer measure form an σ-ideal, it is very natural to consider the following question.…”

“…. In this article, we continue with the study (which was initiated in [1]) of the outer measures λ f on the real line R defined by weak selections f . One of the main results is to show that CH is equivalent to the existence of a weak selection f for which:…”