Spaces of $\sigma $-finite linear measure

Volume 133 / 2013

Ihor Stasyuk, Edward D. Tymchatyn Colloquium Mathematicum 133 (2013), 245-252 MSC: Primary 28A75, 28A78; Secondary 54F50. DOI: 10.4064/cm133-2-11


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 $\sigma $-finite linear measure then $X$ admits a $\sigma $-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 $\sigma $-finite linear measure. In particular, it can be embedded in the $1$-dimensional Nöbeling space $\nu _1^3$ so that the image has $\sigma $-finite linear measure with respect to the usual metric on $\nu _1^3$.


  • Ihor StasyukDepartment of Computer Science
    and Mathematics
    Nipissing University
    100 College Drive, Box 5002
    North Bay, ON, P1B 8L7, Canada
  • Edward D. TymchatynDepartment of Mathematics and Statistics
    University of Saskatchewan
    McLean Hall
    106 Wiggins Road
    Saskatoon, SK, S7N 5E6, Canada

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