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.