Let \({\mathcal R}\) be a ring of subsets of a set X such that \(X\in {\mathcal R}_{\sigma}\), let G be a complete lattice ordered group and let \(f: {\mathcal R}\to G\) be non-negative, bounded and \(\sigma\)-additive (with respect to the order). Following a classical pattern, the authors first extend f to a \(\sigma\)-additive function \(g: {\mathcal R}_{\sigma}\to G.\) Next they extend g to an outer measure like function \(h: {\mathcal P}(X)\to G\) and show that the Carathéodory field \({\mathcal M}\) determined by h contains \({\mathcal R}_{\sigma}\). They ask, among other questions, when h is \(\sigma\)-subadditive on \({\mathcal P}(X)\) and \({\mathcal M}\) is a \(\sigma\)- field. \{For an answer and further references see \textit{J. D. M. Wright} [Meas. Theory, Proc. Conf. Oberwolfach 1975, Lecture Notes Math. 541, 267-276 (1976; Zbl 0357.28011)]. The subject has been also treated by several mathematicians from Bratislava in papers published mainly in Math. Slovaca.\}