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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-11,
author = {Ihor Stasyuk and Edward D. Tymchatyn},
title = {Spaces of $\sigma$-finite linear measure},
journal = {Colloquium Mathematicae},
volume = {131},
year = {2013},
pages = {245-252},
zbl = {1286.28005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-11}
}
Ihor Stasyuk; Edward D. Tymchatyn. Spaces of σ-finite linear measure. Colloquium Mathematicae, Tome 131 (2013) pp. 245-252. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-11/