Intervals of finitely additive set functions (Q919473)

Let \({\mathcal F}\) be an algebra of subsets of a nonempty set \(\Omega\) and let a(\({\mathcal F})\) denote the collection of all additive set functions \({\mathcal F}\to {\mathbb{R}}.\) Consider a set function \(\alpha:{\mathcal F}\to 2^{{\mathbb{R}}}\) such that the set \(\cup_{A\in {\mathcal F}}\alpha (A)\) is bounded, and an order interval \([\xi_ 1,\xi_ 2]\) of a(\({\mathcal F})\) such that the integrals \(\int_{\Omega}\alpha \xi_ 1\) and \(\int_{\Omega}\alpha \xi_ 2\) exist in the sense of the author's earlier paper [Int. J. Math. Math. Sci. 7, 209-233 (1984; Zbl 0593.28001)]. It is shown that \((T_{\alpha}\xi)(A):=\int_{A}\alpha \xi\) exists for all \(\xi \in [\xi_ 1,\xi_ 2]\) and \(A\in {\mathcal F},\) and several properties of the map \(T_{\alpha}:[\xi_ 1,\xi_ 2]\to a({\mathcal F})\) are studied.