Notes on Jordan fields based on an article by Maharam
Let \({\mathcal B}\) be an algebra of subsets of a set X and let \(\mu: {\mathcal B}\to [0,\infty]\) be additive and satisfy \(\mu (A)=\sup \{\mu (A'):\;A\supset A'\in {\mathcal B}\text{ and } \mu (A')<\infty \}.\) The author is concerned with the Jordan field \(\hat {\mathcal B}^{\mu}\) associated with \(\mu\) and with \(\mu\)-integrability, as defined by \textit{D. Maharam} [J. Math. Anal. Appl. 133, No.1, 163-194 (1988; Zbl 0667.28001)]. Among other things, he discusses the question of when \(\hat {\mathcal B}^{\mu}\) is a \(\sigma\)-algebra and the question of the completeness of \(L^ 1(X,\hat {\mathcal B}^{\mu},\mu).\) The discussion involves the topology on X generated by \({\mathcal B}\) as well as the Stone space of \({\mathcal B}.\) \{Reviewer's remarks: (1) Propositions 2-1, 2-6 and 5-2 are invalid. They fail for \(X={\mathbb{N}}\) (with the discrete topology) and \({\mathcal B}=2^ X\), with \(\mu\) chosen suitably. (2) For related material see \textit{S. Gangopadhyay} [Colloq. Math. 58, No.2, 291-300 (1990; Zbl 0719.28002)], and \textit{G. H. Greco} [Ann. Mat. Pura Appl., IV. Ser. 132, 243-255 (1982; Zbl 0553.28005)].\}
scientific article; zbMATH DE number 4192052
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