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Abstract

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.