Nuclear subspace of \(L^ 0\) and the kernel of a linear measure
Let E be a locally convex (separated) topological vector space (over \({\mathbb{R}}\), say) and \(L:E\to L^ 0(\Omega,{\mathcal F},P)\) be a continuous linear functional; L induces a cylindrical probability measure \(\mu\) on E'. The paper studies several questions relating L, L(E) and \(\mu\). One theorem is as follows: if E is an LF space then \(\mu\) is \(\sigma\)- additive iff \(x_ n\to 0\) in E implies that \(L(x_ n)\to 0\) P-a.s. Other theorems relate the nuclearity of E with the \(\sigma\)-additivity of \(\mu\) or the nuclearity of E' with a cylindrical measure \(\nu\) on E.
scientific article; zbMATH DE number 4017987