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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)]}.

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