Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki. "Denseness and Borel complexity of some sets of vector measures." Studia Mathematica 165.2 (2004): 111-124. <http://eudml.org/doc/284601>.

@article{ZbigniewLipecki2004,
abstract = {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.},
author = {Zbigniew Lipecki},
journal = {Studia Mathematica},
keywords = {positive measure; Banach space; vector measure; variation; relatively compact range; Banach space of vector measures; dense set; closed set; -set; -set},
language = {eng},
number = {2},
pages = {111-124},
title = {Denseness and Borel complexity of some sets of vector measures},
url = {http://eudml.org/doc/284601},
volume = {165},
year = {2004},
}

TY - JOUR
AU - Zbigniew Lipecki
TI - Denseness and Borel complexity of some sets of vector measures
JO - Studia Mathematica
PY - 2004
VL - 165
IS - 2
SP - 111
EP - 124
AB - 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.
LA - eng
KW - positive measure; Banach space; vector measure; variation; relatively compact range; Banach space of vector measures; dense set; closed set; -set; -set
UR - http://eudml.org/doc/284601
ER -