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Sigma-ideals and outer measures on the real line

Garcia-Ferreira Salvador, Tomita Artur H., Ortiz-Castillo Yasser Ferman

Journal of Applied Analysis

2019

Vol. 25, nr 1

25--36

A weak selection on ℝ is a function f : [ℝ]2 → ℝ such that f({x, y}) ∈ {x, y} for each {x, y} ∈ [ℝ]2. In this article, we continue with the study (which was initiated in [1]) of the outer measures λf 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 λf (A) = 0 whenever |A| ≤ ω and λf (A) = ∞ otherwise. Some conditions are given for a σ-ideal of ℝ in order to be exactly the family Nf of λf -null subsets for some weak selection f. It is shown that there are 2c pairwise distinct ideals on ℝ of the form Nf , where f is a weak selection. Also, we prove that the Martin axiom implies the existence of a weak selection f such that Nf is exactly the σ-ideal of meager subsets of ℝ. Finally, we shall study pairs of weak selections which are “almost equal” but they have different families of λf -measurable sets.