Real Analysis Exchange

Outer Measures on the Real Line by Weak Selections

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.

Article information

Source
Real Anal. Exchange, Volume 39, Number 1 (2013), 101-116.

Dates
First available in Project Euclid: 1 July 2014