The dual space of \({\mathcal L}^ 1(\mu)\) for a vector measure \(\mu\)
Let \(X\) be a Banach space and let \(\mu\) be an \(X\)-valued measure defined on a \(\sigma\)-algebra \(\Sigma\). The space \({\mathcal L}^ 1(\mu)\) of scalar- valued, \(\mu\)-integrable functions and its topological dual \({\mathcal L}^ 1(\mu)'\) are investigated. The author corrects an earlier characterization of \({\mathcal L}^ 1(\mu)'\) due to \textit{L. Egghe} [Rev. Roum. Math. Pures Appl. 29, 467-471 (1984; Zbl 0555.46015)]. \({\mathcal L}^ 1(\mu)'\) is now represented as the space of scalar measures \(\nu\) that are absolutely continuous with respect to \(\mu\) and satisfy the inclusion \({\mathcal L}^ 1(\mu)\subseteq {\mathcal L}^ 1(\nu)\). A concrete representation of each member of \({\mathcal L}^ 1(\mu)'\) is also given. Basically scattered vector measures with a bounded basis constant are characterized as those measues for which \({\mathcal L}^ 1(\mu)'\) consists of \(\{x'\mu: x'\in X'\}\). The weak convergence in \({\mathcal L}^ 1(\mu)\) is also investigated.
scientific article; zbMATH DE number 480463