×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] FREMLIN D. H.: A direct proof of the Mathes-Wright integral extension theorem. J. London Math. Soc. (2) 11, 1975, 276-284. · Zbl 0313.06016
[2] KLUVÁNEK I.: Note on the extension of measure (Slovak). Mat. fyz. čas. SAV 7, 2. 1957, 108-115.
[3] KLUVÁNEK I.: The extension and closure of vector measure. In Vector and operator valued measures and applications. Academic Press 1975.
[4] POTOCKÝ R.: On random variables having values in vector lattice. Math. Slov. 27, 1977, 267-276.
[5] RIEČAN B.: On the lattice group valued measure. Čas. pro pěst. mat. 101, 1976, 343-349.
[6] ŠIPOŠ J.: Extension of partially ordered group valued measure-like set functions. Čas. pro pěst. mat. 108 (2), 1983, 113-121. · Zbl 0532.28010
[7] VOLAUF P.: Extension and regularity of l-group valued measures. Math. Slov. 27, 1977, No. 1, 47-53. · Zbl 0348.28012
[8] VONKOMEROVÁ M.: On the extension of positive operators. Math. Slovaca, 31, 1981, No. 3. 251-262. · Zbl 0457.46004
[9] WRIGHT J. D. M.: The measure extension problem for vector lattices. Annales de ľInstitut Fourier (Grenoble) 21. 1971. 65-85. · Zbl 0215.48101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.