Spaces of σ-finite linear measure

• Autorzy: Ihor Stasyuk , Edward D. Tymchatyn
• Języki publikacji: EN

Abstrakty

EN

Spaces of finite n-dimensional Hausdorff measure are an important generalization of n-dimensional polyhedra. Continua of finite linear measure (also called continua of finite length) were first characterized by Eilenberg in 1938. It is well-known that the property of having finite linear measure is not preserved under finite unions of closed sets. Mauldin proved that if X is a compact metric space which is the union of finitely many closed sets each of which admits a σ-finite linear measure then X admits a σ-finite linear measure. We answer in the strongest possible way a 1989 question (private communication) of Mauldin. We prove that if a separable metric space is a countable union of closed subspaces each of which admits finite linear measure then it admits σ-finite linear measure. In particular, it can be embedded in the 1-dimensional Nöbeling space ν₁³ so that the image has σ-finite linear measure with respect to the usual metric on ν₁³.

Kategorie tematyczne

• 28A75: Length, area, volume, other geometric measure theory
• 54F50: Spaces of dimension ≤ 1 ; curves, dendrites
• 28A78: Hausdorff and packing measures

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2013
• Tom: 133
• Numer: 2
• Strony: 245-252

Daty

wydano

2013

Twórcy

autor

Ihor Stasyuk

• Department of Computer Science and Mathematics, Nipissing University, 100 College Drive, Box 5002, North Bay, ON, P1B 8L7, Canada

autor

Edward D. Tymchatyn

• Department of Mathematics and Statistics, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada

Identyfikatory

DOI

10.4064/cm133-2-11