基于Steiner树的连通集的外线性测度
Outer linear measure of connected sets via Steiner trees
摘要：
我们恢复了度量连续统的线性测度在Steiner树中的一个旧定义，独立地归因于Menger（1930）和Choquet（1938）。我们将其推广到任何度量空间，并提供了一个鲜为人知的Choquet定理的证明，即它与任何连通度量空间的外线性测度相吻合。作为推论，我们得到了关于连续统的线性测度的下半连续性的戈戈布定理（1928）和关于集合闭包的线性测度的博格纳定理（1989）的简单证明。除了外线性测度的定义外，我们不使用任何测度论。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
