On vector measures with separable range (Q869254)

Ranges of vector measures have interesting properties. Indeed, let \(X\) be a Banach space, \(\Omega\) an abstract set, and \(\Sigma\) a \(\sigma\)-algebra of subsets of \(\Omega\). If \(\nu:\Sigma\to X\) is countably additive measure, then it was shown by \textit{R. G. Bartle, N. Dunford} and \textit{J. Schwartz} [Can. J. Math. 7, 289-305 (1955; Zbl 0068.09301)] that the range of \(\nu\), \(\nu(\Sigma)=\{\nu(A):A\in\Sigma\}\), is a relatively weakly compact subset of \(X\). The paper under review is a contribution the study of the norm separability of ranges of vector measures; conditions are given which guarantee that \(X^*\)-valued measures with \(\sigma\)-finite variation have norm separable range. More precisely, it is shown that if \(X\) is weakly Lindelöf determined (i.e.\ \((B_{X^*},\omega^*)\) is homeomorphic to a subset of countably supported elements of a product space \([-1,1]^\Lambda\)), then the above property is equivalent to the fact that every Radon probability measure on \((B_{X^*},w^*)\) having separable support (here \(w^*\) stands for the topology \(\sigma(X^*,X)\)). It should be noted that weakly compactly generated Banach spaces are weakly Lindelöf determined. An example of a measure with values in the space \(\ell^\infty(\omega_1)\) having finite variation and non norm separable range is constructed. Hence, if \(X\) is isomorphic to a subspace of \(\ell^1(\omega_1)\), then there exists an \(X^*\)-valued measure with the above properties.