2013/2014 Outer Measures on the Real Line by Weak Selections
J. A. Astorga-Moreno, S. Garcia-Ferreira
Real Anal. Exchange 39(1): 101-116 (2013/2014).

## Abstract

A weak selection on an infinite set $$X$$ is a function $$f:[X]^2 \to X$$ such that $$f(F) \in F$$ for each $$F \in [X]^2 := \{ E \subseteq X : |E| = 2 \}$$. If $$f: [X]^2 \to X$$ is a weak selection and $$x, y \in \mathbb{R}$$, then we say that $$x \lt_f y$$ if $$f(\{x, y\}) = x$$ and $$x \leq_f y$$ if either $$x = y$$ or $$x \lt_f y$$. Given a weak selection $$f$$ on $$X$$ and $$x, y \in X$$, we let $$(x,y]_f = \{z \in X : x \lt_f z \le_f y \}$$. If $$f: [\mathbb{R}]^2 \to \mathbb{R}$$ is a weak selection and $$A \subseteq \mathbb{R}$$, then we define $\lambda^{*}_{f}(A):=\inf\Big\{\sum_{n \in \mathbb{N}} |b_{n} - a_{n}| \, : \, A \subseteq \bigcup_{n \in \mathbb{N}}(a_{n},b_{n}]_{f} \Big\}$ if there exists a countable cover by semi open $$f$$-intervals of $$A,$$ and if there is not such a cover, then we say that $$\lambda^{*}_{f}(A)=+\infty$$. This function $$\lambda^{*}_{f}\:mathcal{P}(\mathbb{R}) \longrightarrow [0,+\infty]$$ is an outer measure on the real line $$\mathbb{R}$$ which generalizes the Lebesgue outer measure. In this paper, we show several interesting properties of these kind of outer measures.

## Citation

J. A. Astorga-Moreno. S. Garcia-Ferreira. "Outer Measures on the Real Line by Weak Selections." Real Anal. Exchange 39 (1) 101 - 116, 2013/2014.

## Information

Published: 2013/2014
First available in Project Euclid: 1 July 2014

zbMATH: 1298.28006
MathSciNet: MR3261902

Subjects:
Primary: 28A12 , 28A99
Secondary: 28B15 , 28E15

Keywords: Lebesgue outer measure , measurable set , null set , outer measure , weak selection

JOURNAL ARTICLE
16 PAGES

Vol.39 • No. 1 • 2013/2014