# Outer linear measure of connected sets via Steiner trees

## TL;DR

This paper revisits and generalizes the concept of linear measure of connected sets using Steiner trees, providing new proofs for classical theorems without relying on measure theory.

## Contribution

It extends the definition of linear measure via Steiner trees to all metric spaces and proves its equivalence with outer linear measure for connected spaces.

## Key findings

- Proves the equivalence of Steiner tree-based measure and outer linear measure in metric spaces.
- Provides simple proofs of classical theorems on linear measure of continua.
- Generalizes the concept of linear measure without measure theory.

## 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 Go{\l}\k{a}b's theorem (1928) on the lower semicontinuity of linear measure of continua and a theorem of Bogn\'ar (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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Source: https://tomesphere.com/paper/1908.02230