2021 Outer linear measure of connected sets via Steiner trees
Konrad J. Swanepoel
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Real Anal. Exchange 46(1): 207-232 (2021). DOI: 10.14321/realanalexch.46.1.0207

Abstract

We resurrect an old definition of the linear measure of a metric continuum in terms of Steiner trees, independently due to Menger (1930) and Choquet (1938). We generalise it to any metric space and provide a proof of a little-known theorem of Choquet that it coincides with the outer linear measure for any connected metric space. As corollaries we obtain simple proofs of Gołąb’s theorem (1928) on the lower semicontinuity of linear measure of continua and a theorem of Bognár (1989) on the linear measure of the closure of a set. We do not use any measure theory apart from the definition of outer linear measure.

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Konrad J. Swanepoel. "Outer linear measure of connected sets via Steiner trees." Real Anal. Exchange 46 (1) 207 - 232, 2021. https://doi.org/10.14321/realanalexch.46.1.0207

Information

Published: 2021
First available in Project Euclid: 14 October 2021

Digital Object Identifier: 10.14321/realanalexch.46.1.0207

Subjects:
Primary: 28A75

Keywords: Bognár’s theorem , Gołąb’s theorem on semicontinuity , outer linear measure , rectifiable continua , Steiner trees

Rights: Copyright © 2021 Michigan State University Press

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Vol.46 • No. 1 • 2021
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