Continuous linear functionals on the space of Borel vector measures

Pola Siwek¹

¹Institute of Mathematics University of Silesia Bankowa 14 40-007 Katowice, Poland

Annales Polonici Mathematici, Tome 93 (2008) no. 3, pp. 199-209

Résumé

We study properties of the space $\mathcal M$ of Borel vector measures on a compact metric space $X$, taking values in a Banach space $E$. The space $\mathcal M$ is equipped with the Fortet–Mourier norm $\|\cdot \|_{\mathcal F}$ and the semivariation norm $\|\cdot \|(X)$. The integral introduced by K. Baron and A. Lasota plays the most important role in the paper. Investigating its properties one can prove that in most cases the space $(\mathcal M, \|\cdot \|_{\mathcal F})^*$ is contained in but not equal to the space $(\mathcal M,\|\cdot \|(X))^*$. We obtain a representation of the continuous functionals on $\mathcal M$ in some particular cases.

Détail

DOI : 10.4064/ap93-3-1

Keywords: study properties space mathcal borel vector measures compact metric space taking values banach space space mathcal equipped fortet mourier norm cdot mathcal semivariation norm cdot integral introduced baron lasota plays important role paper investigating its properties prove cases space mathcal cdot mathcal * contained equal space mathcal cdot * obtain representation continuous functionals mathcal particular cases