𝜎-ideals and outer measures on the real line

Abstract

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