On the extension of a vector-submeasure (Q1061882)

This paper discusses an extension of a vector submeasure \(\eta\) ; it has been shown that if \(\eta\), in addition to being defined on a ring of sets \({\mathfrak R}\) and taking values in an L-normed Banach lattice, be order bounded, exhausting and continuous from above, then it can be extended to a \(\sigma\)-ring \({\mathfrak R}_ 0\supseteq {\mathfrak R}\) so as to ensure that the extended submeasure \({\tilde \eta}\) inherits the properties of \(\eta\). Moreover, with respect to the topology generated by \({\tilde \eta}\) on \({\mathfrak R}_ 0\), \({\mathfrak R}\) is dense in \({\mathfrak R}_ 0\).