Compactness in spaces of group-valued contents, the Vitali-Hahn-Saks theorem and Nikodým’s boundedness theorem.

*(English)*Zbl 0604.28006Relatively compact subsets of ca(R,G) in the topology of pointwise convergence are characterized with ca(R,G) being the space of all (\(\sigma\)-additive) measures on a \(\sigma\)-complete Boolean ring R and values in a topological group G; the proof of the characterization is carried through such that it yields, without extra work, the Vitali-Hahn- Saks theorem for s-bounded (finitely additive) contents, Nikodým’s boundedness theorem (for contents with values in quasi-normed groups) and Rosenthal’s lemma. For a compactness criterion in the space sa(R,G) of all G-valued s-bounded contents on a Boolean ring R, the notion of quasi- uniform s-boundedness is introduced. For different notions of boundedness there are different versions of Nikodým’s theorem in literature. Here the different notions of boundedness and the connections between them are studied. This study together with the version of Nikodým’s theorem for quasi-normed groups immediately yields the finitely additive generalization of C. Constantinescu’s version [Libertas Math. 1, 51-73 (1981; Zbl 0482.28009)]. All results are also discussed in case of R being not \(\sigma\)-complete and G not a topological group.

##### MSC:

28B10 | Group- or semigroup-valued set functions, measures and integrals |