On measures integrating all functions of a given vector lattice
Let E be a vector lattice of real-valued functions defined on a set X, and let \({\mathcal M}(E)\) be the family of all (non-negative, countably additive) measures \(\mu\) on \(\sigma\) (E) such that \(E\subset {\mathcal L}_ 1(X,\sigma (E),\mu).\) Here \(\sigma\) (E) denotes the smallest \(\sigma\)- algebra in X making all functions in E measurable. This paper is concerned with the statement that every \(\mu \in {\mathcal M}(E)\) is \(\tau\)- smooth with respect to the lattice \({\mathcal H}(E):=\{\{f\geq 1\}: f\in E\},\) i.e. \(\lim_{\alpha}\mu (H_{\alpha})=0\) for each net \((H_{\alpha})\) in \({\mathcal H}(E)\) with \(H_{\alpha}\downarrow \emptyset.\) Under some additional assumptions on E, this statement can be described by smoothness properties of the nonnegative linear functionals on E as well as by closure properties of \({\mathcal H}(E)\)-ultrafilters. An application of this result to various topological situations leads to some new measure- theoretic characterizations of realcompact, Borel-complete, and N-compact spaces, respectively.
scientific article; zbMATH DE number 4010837