Outer linear measure of connected sets via Steiner trees

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.