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On extension of group valued measures. (English) Zbl 0760.28007
Let \(G\) be a monotone \(\sigma\)-complete partially ordered group with the property that for every nonzero \(x\in G\) there exists a \(\sigma\)-order continuous positive homomorphism \(f: G\to\mathbb{R}\) (\(f\in G^ <_ +\) in the author’s notation) with \(f(x)\neq 0\). Let \(\mathcal A\) be an algebra of subsets of a set \(X\) and let \(\mu: {\mathcal A}\to G\) be a measure, i.e., \(\mu\) is additive and \(\mu(A_ n)\searrow 0\) for every decreasing sequence \((A_ n)\) in \(\mathcal A\) with empty intersection. The author is concerned with extending \(\mu\) to a complete \(G\)-valued measure on a \(\sigma\)-algebra \({\mathcal S}_ \mu({\mathcal A})\) of subsets of \(X\) containing \(\mathcal A\). The proof applies the classical case \(G=\mathbb{R}\) along with I. Kluvánek’s approach [Mat.-Fyz. Čas., Slovensk Akad. Vied. 7, 108-115 (1957)] to that case.
{Reviewer’s remarks: (1) The formula for \({\mathcal S}_ \mu({\mathcal A})\) claimed in the proof of Lemma 15 is invalid. Indeed, let \(\mathcal A\) be the algebra of finite and cofinite subsets of a set \(X\) with \(\text{card }X=2^{\aleph_ 0}\), let \(G=\mathbb{R}^ X\), and let \(\mu(A)=1_ A\) for \(A\in{\mathcal A}\). Then \({\mathcal S}_ \mu({\mathcal A})\) is the \(\sigma\)-algebra of countable and cocountable subsets of \(X\) while \({\mathcal S}_{f\circ\mu}({\mathcal A})=2^ X\) for each \(f\in G^ <_ +\); cf. G. W. Mackey [Bull. Am. Math. Soc. 50, 719-722 (1944; Zbl 0060.134)]. (2) The author’s setting is quite similar to that of Z. Riečanová and I. Rosová [Math. Nachr. 106, 201-209 (1982; Zbl 0507.28007)]}.

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