Quasi-Radon measures
Let \(X\) be a complete regular topological space, \(\mathcal K\) (resp. \(\Sigma\)) the family of closed (resp. compact) subsets of \(X\), \(M(X)\) (resp. \(C_ b(X)\)) the space of Borel (resp. bounded continuous) real valued functions on \(X\), \(Y\) a lattice-normed space by means of the Riesz space \(F\) [ cf. \textit{L. V. Kantorovich}, \textit{B. Z. Vulich} and \textit{A. G. Pinsker}, ``Functional analysis in partially ordered spaces'' (1950; Zbl 0037.072)] with generalized norm \(| y|\in F\), whenever \(y\in Y\). By \(F-ba({\mathcal A},Y)\) (resp. \(F-bca({\mathcal A},Y))\) it is denoted the set of all finitely additive (resp. \(\sigma\)-additive) measures \(\mu: {\mathcal A}\to Y\) with finite \(F\)-variation \(|\mu|\) on a given algebra \(\mathcal A\) of subsets of \(X\). A finitely additive measure \(\mu: {\mathcal A}\to Y\) is called Radon (resp. quasi-Radon) measure on \(\mathcal A\) if and only if \(\mu(E)=o-\lim\{\mu(K): K\in\Sigma_ E\}\), for every \(E\in{\mathcal A}\) (resp. \(E\in\Sigma\cap\mathcal A)\) with \(\Sigma_ E:=\{K: K\in\Sigma\cap {\mathcal A}, K\subseteq E\}\). Further \(\mu\) is called regular (resp. quasi-regular) measure if \(\mu(E)=o-\lim\{\mu(F): F\in K_ E\}\) for every \(E\in{\mathcal A}\) (resp. \(E\in {\mathcal K}\cap{\mathcal A})\) with \(K_ E:=\{F: F\in{\mathcal K}\cap {\mathcal A}, F\subseteq E\}\). The author showed that if \(\mu\in F-ba({\mathcal A},Y)\) and either (i) \({\mathcal A}=\alpha({\mathcal K}\cap{\mathcal A})\) (the algebra generated by the class \({\mathcal K}\cap{\mathcal A})\) or (ii) \({\mathcal A}=\sigma({\mathcal K}\cap{\mathcal A})\) (the \(\sigma\)-algebra generated by the class \({\mathcal K}\cap{\mathcal A})\) and \(\mu\in F-bca({\mathcal A},Y)\), then \(\mu\) is a quasi-regular (resp. quasi-Radon) measure if and only if \(|\mu|\) is a quasi-regular (resp. quasi-Radon) measure. Moreover, the bounded quasi-Radon measure \(\mu\) on the algebra \(\alpha({\mathcal K}\cap{\mathcal A})\) is \(\sigma\)-additive on \(\mathcal A\) if \(\mu\in F-ba({\mathcal A},Y)\). Similarly, if \(\mu\) is a Radon measure in \(F-ba({\mathcal A},Y)\) then \(\mu\) is \(\sigma\)-additive on \(\mathcal A\). Following the reviewer [Can. J. Math. 35, 353-372 (1983; Zbl 0487.28010); Math. Ann. 287, No. 2, 335-341 (1990; Zbl 0677.46029); J. Aust. Math. Soc., Ser. A 51, No. 2, 187-215 (1991; Zbl 0753.28010)] and \textit{J. D. M. Wright} [Proc. Lond. Math. Soc., III. Ser. 19, 107-122 (1969; Zbl 0186.465)] the author defines the space \(L^ 1(\mu)\) of \(\mu\)-integrable real-valued functions in \(M(X)\) with respect to a measure \(\mu\) in \(F- ba({\mathcal A},Y)\) and he obtains an integral representation theorem concerning a class of linear operators \(T: {\mathfrak L}\to Y\) with \({\mathfrak L}\) a vector sublattice of \(C_ b(X)\) with \(1\in{\mathfrak L}\). Besides given a quasi-Radon measure \(\mu_ 0\) in \(F-ba({\mathcal A},Y)\), where \(\mathcal A\) denotes a certain algebra of subsets of \(X\), there exists exactly one extension \(\mu\) of \(\mu_ 0\) in \(F-ba({\mathcal B},Y)\) and \(\mu\) is a quasi-Radon measure on the Borel algebra \(\mathcal B\) of subsets of \(X\). The work also contains interesting applications of the derived results on the classical moment problem, stated in a suitable vectorial version. In the reviewer's opinion the author has written a nice paper in the setting of measures and integrals with values in partially ordered vector spaces.
scientific article; zbMATH DE number 56736
scientific article