Extension of quasitriangular submeasure (Q1311171)

Let \(\Sigma\) be an \(m\)-class, i.e., a nonempty class of sets closed under difference (a notion slightly more general than that of a ring of sets). Moreover, let \(\varphi\) be a quasi-triangular submeasure on \(\Sigma\), i.e., \(\varphi\) takes values in \([0, \infty]\), is monotone, vanishes at \(\emptyset\) and satisfies the condition that, given \(\varepsilon>0\) there exists \(\delta>0\) such that \(\varphi (A \cup B) < \varepsilon\) whenever \(A,B \in \Sigma\) and \(\varphi (A), \varphi (B)<\delta\). \{This condition is due to \textit{I. Dobrakov} (1974).\} The authors prove that if \(\varphi\) is additionally upper continuous at \(\emptyset\) and exhaustive, then it extends to a quasi-triangular submeasure \(\overline\varphi\) on a \(\sigma\)-ring \(\overline \Sigma\) containing \(\Sigma\) which is continuous at \(\emptyset\). The case where \(\varphi\) is subadditive is due independently to \textit{V. N. Aleksyuk} and \textit{F. D. Beznosikov} (1972) and \textit{L. Drewnowski} (1972). The authors' proof is a modification of Drewnowski's. Uniqueness is not discussed; nor are relations to Dobrakov's extension theorem concerning submeasures intermediate between quasi-triangular and subadditive ones. For a proof of that theorem see \textit{L. Drewnowski} [Colloq. Math. 38, 243-253 (1978; Zbl 0398.28003)].