Spaces of $\sigma $-finite linear measure

Ihor Stasyuk ¹   ; Edward D. Tymchatyn ²

¹ Department of Computer Science and Mathematics Nipissing University 100 College Drive, Box 5002 North Bay, ON, P1B 8L7, Canada
² Department of Mathematics and Statistics University of Saskatchewan McLean Hall 106 Wiggins Road Saskatoon, SK, S7N 5E6, Canada

Colloquium Mathematicum, Tome 133 (2013) no. 2, pp. 245-252

Résumé

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 $\nu _1^3$.

Détail

DOI : 10.4064/cm133-2-11

Keywords: spaces finite n dimensional hausdorff measure important generalization n dimensional polyhedra continua finite linear measure called continua finite length first characterized eilenberg well known property having finite linear measure preserved under finite unions closed sets mauldin proved compact metric space which union finitely many closed sets each which admits sigma finite linear measure admits sigma finite linear measure answer strongest possible question private communication mauldin prove separable metric space countable union closed subspaces each which admits finite linear measure admits sigma finite linear measure particular embedded dimensional beling space image has sigma finite linear measure respect usual metric