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Outer linear measure of connected sets via Steiner trees

Swanepoel, Konrad ORCID: 0000-0002-1668-887X (2020) Outer linear measure of connected sets via Steiner trees. Real Analysis Exchange, 46 (1). ISSN 0147-1937 (In Press)

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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.

Item Type: Article
Official URL: https://projecteuclid.org/journals/real-analysis-e...
Additional Information: © 2020 Michigan State University Press
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited: 09 Oct 2020 13:18
Last Modified: 04 Sep 2021 23:18
URI: http://eprints.lse.ac.uk/id/eprint/106741

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