On the extensions of a family of vector submeasures (Q1082464)

The authors study submeasures with values in an A(bstract) L-space, i.e. a Banach lattice with the additional property that the norm is additive on the positive cone. One would expect such a submeasure to be defined as a monotone, subadditive map \(\mu\) on a ring into the vector lattice vanishing at the empty set. Here however the subadditivity of \(\mu\) is replaced by that of \(\| \mu (\cdot)\|\). This latter real-valued set function is an ordinary submeasure in the sense of \textit{L. Drewnowski} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 20, 269-276 (1972; Zbl 0249.28004.), ibid. 277-286 (1972; Zbl 0249.28005.)]. The authors re-prove Drewnowski's exhaustion principle [op. cit., part II, 4.7] for this particular real submeasure. The major part of the paper is the proof that one can simultaneously extend each of a family of c.(continuous) f.(rom) \(a.(bove)(=sequentially\) ordercontinuous) submeasures on a sigma-algebra to the whole power set of the underlying space, preserving the order \(\mu \leq \nu\) if and only if \(\| \mu (\cdot)\| \leq \| \nu (\cdot)\|\). The extension \({\bar \mu}\) is such that for each A there is a B in the original sigma-algebra with \({\bar \mu}(A\Delta B)=0\). This result is combined with a result of a previous paper by the authors [Indian J. Pure Appl. Math. 15, 731-748 (1984; Zbl 0571.28005.)] to show that one may extend c.f.a. exhaustive submeasures from a ring to the entire power set in such a way that the ordering is preserved and the ring is dense in the power set with respect to the FN-topology induced by all of the extended submeasures. There are a number of misprints.