## Spaces of $\sigma $-finite linear measure

### Volume 133 / 2013

#### Abstract

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 $\nu _1^3$.