Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki

Studia Mathematica, Tome 162 (2004), p. 111-124 / Harvested from The Polish Digital Mathematics Library

Résumé

Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets $_{ν}(X)$ and $_{ν}(X)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient conditions that $_{ν}(X)$ [resp., $_{ν}(X)$] be dense in ca(Σ,ν,X) [resp., ca(Σ,X)]. We also show that $_{ν}(X)$ and $_{ν}(X)$ are always $G_{δ}$-sets and establish necessary and sufficient conditions that they be $F_{σ}$-sets in the respective spaces.

Détail

Publié le : 2004-01-01

Zbl 1055.28006

EUDML-ID : urn:eudml:doc:284601