Denseness and Borel complexity of some sets of vector measures

• Autorzy: Zbigniew Lipecki
• Języki publikacji: EN

Abstrakty

EN

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 $𝓔_{ν}(X)$ and $𝒜_{ν}(X)$ 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 $𝓔_{ν}(X)$ [resp., $𝒜_{ν}(X)$] be dense in ca(Σ,ν,X) [resp., ca(Σ,X)]. We also show that $𝓔_{ν}(X)$ and $𝒜_{ν}(X)$ are always $G_{δ}$-sets and establish necessary and sufficient conditions that they be $F_{σ}$-sets in the respective spaces.

Kategorie tematyczne

• 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
• 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
• 28B05: Vector-valued set functions, measures and integrals
• 46E27: Spaces of measures
• 46G10: Vector-valued measures and integration

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2004
• Tom: 165
• Numer: 2
• Strony: 111-124

Daty

wydano

2004

Twórcy

autor

Zbigniew Lipecki

• Institute of Mathematics, Polish Academy of Sciences, Wrocław Branch, Kopernika 18, 51-617 Wrocław, Poland

Identyfikatory

DOI

10.4064/sm165-2-2