# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Denseness and Borel complexity of some sets of vector measures

### Tom 165 / 2004

Studia Mathematica 165 (2004), 111-124 MSC: Primary 28B05, 46G10, 46E27, 54H05; Secondary 46A16. DOI: 10.4064/sm165-2-2

#### Streszczenie

Let $\nu$ be a positive measure on a $\sigma$-algebra ${\mit \Sigma }$ of subsets of some set and let $X$ be a Banach space. Denote by $\mathop {\rm ca}\nolimits ({\mit \Sigma },X)$ the Banach space of $X$-valued measures on ${\mit \Sigma }$, equipped with the uniform norm, and by $\mathop {\rm ca}\nolimits ({\mit \Sigma },\nu ,X)$ its closed subspace consisting of those measures which vanish at every $\nu$-null set. We are concerned with the subsets ${\mathcal E}_\nu (X)$ and ${\mathcal A}_\nu (X)$ of $\mathop {\rm ca}\nolimits ({\mit \Sigma },X)$ defined by the conditions $|\varphi |=\nu$ and $|\varphi |\geq \nu$, respectively, where $|\varphi |$ stands for the variation of $\varphi \in \mathop {\rm ca}\nolimits ({\mit \Sigma },X)$. We establish necessary and sufficient conditions that ${\mathcal E}_\nu (X)$ [resp., ${\mathcal A}_\nu (X)$] be dense in $\mathop {\rm ca}\nolimits ({\mit \Sigma },\nu ,X)$ [resp., $\mathop {\rm ca}\nolimits ({\mit \Sigma },X)$]. We also show that ${\mathcal E}_\nu (X)$ and ${\mathcal A}_\nu (X)$ are always $G_\delta$-sets and establish necessary and sufficient conditions that they be $F_\sigma$-sets in the respective spaces.

#### Autorzy

• Zbigniew LipeckiInstitute of Mathematics
Polish Academy of Sciences
Wroc/law Branch
Kopernika 18
51-617 Wroc/law, Poland
e-mail

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