$sigma$-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.