On extensions of \(\sigma\)-additive set functions with values in a topological group (Q1085296)

The authors discuss two types of extensions of a \(\sigma\)-additive set function defined on a \(\sigma\)-algebra \({\mathfrak M}\) of subsets of a set X with values in an Abelian topological semi-group with neutral element or an Abelian complete Hausdorff topological group. These extensions are Peano-Jordan completions and a Łoś-Marczewski extension to the \(\sigma\)-algebra generated by \({\mathfrak M}\) and one extra subset of X. The discussion is based upon the reviewer's paper in Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 22, 19-27 (1974; Zbl 0275.28013). The only result which goes essentially beyond that paper is Theorem 3.1 concerning extensions of the second type. \{Reviewer's remarks: (1) Peano-Jordan completions were also studied in another paper by the reviewer [Glas. Math., III. Ser. 18(38), 87-90 (1983; Zbl 0597.28014)]. (2) Theorem 3.1 follows from a result of the reviewer [Bull. Acad. Pol. Sci., Sér. Sci. Math. 28, 441-445 (1980; Zbl 0469.28009)]. (3) Lemma 3.1 is trivial, as, in its setting, \(\sigma\)- additivity implies exhaustivity. (4) The question formulated at the end of Section 1 has a negative answer due to Banach, Kuratowski and Ulam (1929-30).\}