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Spaces of σ-finite linear measure

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Stasyuk I., Tymchatyn E.

Colloquium Mathematicum

2013

tom 133

nr 2

245-252

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Stasyuk I.

1 Tymchatyn E.

1 2013

Wyniki wyszukiwania

Spaces of σ-finite linear measure