𝜎-ideals and outer measures on the real line

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