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

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: $$\mathcal \lambda_f(A)= \begin{cases} 0 & \text{if $|A| \leq \omega$,}\\ \infty & \text{otherwise.} \end{cases}$$ 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.