Existence and integral representation of regular extensions of measures
Let \(X\) be a nonempty set, let \({\mathcal L}\) be a lattice of subsets of \(X\) with \(\emptyset\in{\mathcal L}\) and let \({\mathcal A}\) stand for the algebra of subsets of \(X\) generated by \({\mathcal L}\). Let, further, \({\mathcal B}\) be a subalgebra of \({\mathcal A}\) and let \(\nu\) be a content on \({\mathcal B}\), i.e., \(\nu:{\mathcal B}\to [0,\infty)\) is additive. Denote by \(E\) [resp., \(E_r\)] the convex set of all [resp., all \({\mathcal L}\)-inner regular] extensions of \(\nu\) to a content on \({\mathcal A}\). The author proves, under a variant of Marczewski's regularity condition on \(\nu\), that given \(\lambda\in E\) [resp., \(\lambda\in \text{ex }E\)] there exists \(\mu\in E_r\) [resp., \(\mu\in \text{ex }E_r\)] with \(\mu(L)\geq \lambda(L)\) for all \(L\in{\mathcal L}\) (Proposition 3.1(a)). Using the second part of this result, he obtains a Choquet type representation theorem for \(E_r\). The paper also contains analogous results in the \(\sigma\)-additive case. In that case \({\mathcal L}\) is assumed to be a \(\delta\)-lattice, \({\mathcal A}\) stands for the \(\sigma\)-algebra generated by \({\mathcal L}\) and the outer extension of \(\nu\) is assumed to be \(\sigma\)-smooth at \(\emptyset\) on \({\mathcal L}\).NEWLINENEWLINENEWLINE\{Reviewer's remarks: (1) In connection with the proof of Lemma 2.1 see also [\textit{J. Lembcke}, Lect. Notes Math. 794, 45-48 (1980; Zbl 0479.28001)]. (2) The assertion of the first part of Proposition 3.1(a) was also established, in a related situation, by the reviewer [Math. Nachr. 146, 167-173 (1990; Zbl 0693.28003), Corollary 1 and Theorem 2]\}.
scientific article; zbMATH DE number 1605287