Swanepoel, Konrad ORCID: 0000-0002-1668-887X
(2021)
*Outer linear measure of connected sets via Steiner trees.*
Real Analysis Exchange, 46 (1).
207 - 232.
ISSN 0147-1937

Text (Outer linear measure of connected sets via Steiner trees)
- Accepted Version
Download (465kB) |

## 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: | © 2021 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: | 09 Nov 2024 00:45 |

URI: | http://eprints.lse.ac.uk/id/eprint/106741 |

### Actions (login required)

View Item |