The space of scalarly integrable functions for a Fréchet-space-valued measure
Let \(X\) be a Fréchet space, \(\nu:\Sigma\to X\) a vector measure and \((\Omega,\Sigma)\) be a measurable space. Consider the space \(L_w^1(\nu)\) of all equivalence classes of scalarly integrable functions \(f:\Omega\to\mathbb R\). Clearly the usual space \(L^1(\nu)\) is a closed subspace of \(L_w^1(\nu)\). When \(X\) is a Banach space, it is known that \(L_w^1(\nu)\) can be completed to a Banach space. In this interesting paper, the authors prove that \(L_w^1(\nu)\) is a complete space, when \(X\) is a metrizable locally convex Hausdorff space (and a Fréchet space when \(X\) is a Fréchet space). This result is important, since \(L^1(\nu)\) fails, in general, to be complete, when \(X\) is simply a metrizable locally convex Hausdorff space. The authors also prove that every Fréchet lattice with the \(\sigma\)-Fatou property and with a weak unit in its \(\sigma\)-order continuous part is isomorphic to \(L_w^1(\nu)\) for a suitable measure \(\nu\).
scientific article; zbMATH DE number 5555221
scientific article