Real Analysis Exchange

Outer Measures on the Real Line by Weak Selections

J. A. Astorga-Moreno and S. Garcia-Ferreira

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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

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

First available in Project Euclid: 1 July 2014

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A12: Contents, measures, outer measures, capacities 28A99: None of the above, but in this section
Secondary: 28B15: Set functions, measures and integrals with values in ordered spaces 28E15: Other connections with logic and set theory

outer measure Lebesgue outer measure measurable set null set weak selection


Astorga-Moreno, J. A.; Garcia-Ferreira, S. Outer Measures on the Real Line by Weak Selections. Real Anal. Exchange 39 (2013), no. 1, 101--116.

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