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.

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.