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- W2945149044 abstract "Abstract A weak selection on {mathbb{R}} is a function {fcolon[mathbb{R}]^{2}tomathbb{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 Arvertleqomega} 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." @default.
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- W2945149044 date "2019-06-01" @default.
- W2945149044 modified "2023-09-30" @default.
- W2945149044 title "𝜎-ideals and outer measures on the real line" @default.
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- W2945149044 doi "https://doi.org/10.1515/jaa-2019-0003" @default.
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