$\sigma$-Ideals and outer measures on the real line

Abstract

A {\it weak selection} on $\mathbb{R}$ is a function $f:[\mathbb{R}{]}^2\to \mathbb{R}$ 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 \cite{ag}) 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: \[ \mathcal λ_f(A)= \begin{cases} 0 & \text{if $|A|\le \omega$,}\\ \infty & \text{otherwise.} \end{cases} \] Some conditions are given for a $\sigma$-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 Martin Axiom implies the existence of a weak selection $f$ such that ${\mathcal{N}}_f$ is exactly the $\sigma$-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.