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On extension of submeasures. (English) Zbl 0596.28002

Let \({\mathcal R}\) be a ring of subsets of a nonempty set T. According to Definition 1 in an earlier paper by the author [Diss. Math. 112, 35 p. (1974; Zbl 0292.28001)] a set function \(\mu:{\mathcal R}\to [0,+\infty)\) is a submeasure if it is monotone, continuous \((A_ n\searrow {\mathcal R}0\Rightarrow \mu (A_ n)\to 0),\) and subadditively continuous (for all \(A\in {\mathcal R}\) and all \(\epsilon >0\) there exists \(\delta >0\) such that \(B\in {\mathcal R},\mu (B)<\delta \Rightarrow \mu (A)-\epsilon \leq \mu (A- B)\leq \mu (A)\leq \mu (A\cup B)\leq \mu (A)+\epsilon).\quad\) The property of being subadditively continuous is equivalent to the following property: \(A,A_ n\in {\mathcal R},n=1,2,...\) and \(\mu (A_ n\Delta A)\to 0\Rightarrow \mu (A_ n)\to \mu (A).\) A set function \(\mu:{\mathcal R}\to [0,+\infty]\) is exhaustive if \(\mu (A_ n)\to 0\) for each infinite sequence \(A_ n\in {\mathcal R},n=1,2,...,\) of pairwise disjoint sets. In Theorem 18 in the above-cited paper [op. cit.], the author has proved that a uniform, subadditive or additive submeasure \(\mu:{\mathcal R}\to [0,+\infty)\) has a unique extension of the same type to \(\sigma\) (\({\mathcal R})\)- the \(\sigma\)-ring generated by \({\mathcal R}\)- if and only if it is exhaustive. Two additional, rather clumsy, conditions were needed to obtain the extension theorem for nonuniform submeasures. In this note, using a more transparent approach he shows that these conditions may be replaced by the following: (i) For each \(\epsilon >0\) there is a \(\delta >0\) such that \(A,B\in {\mathcal R},\) \(\mu (A),\mu (B)\leq \delta\) implies \(\mu (A\cup B)<\epsilon\) (pseudometric generating property) and \((ii)\quad A_ n\in {\mathcal R},n=1,2,...,\) and \(\mu (A_ n\Delta A_ m)\to 0\) as \(n,m\to \infty\) implies that \(\mu (A_ n)-\mu (A_ m)\to 0\) as \(n,m\to \infty.\)

MSC:

28A12 Contents, measures, outer measures, capacities
28A10 Real- or complex-valued set functions

Citations:

Zbl 0292.28001
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References:

[1] DOBRAKOV L.: On submeasures I. Dissertationes Math. 112, Warszawa 1974, 1-35. · Zbl 0292.28001
[2] DOBRAKOV I., FARKOVÁ J.: On submeasures II. Math. Slovaca 30, 1980, 65-81. · Zbl 0428.28001
[3] DREWNOWSKI L.: On the continuity of certain non-additive set functions. Colloquium Math. 38, 1978, 243-253. · Zbl 0398.28003
[4] DREWNOWSKI L.: On complete submeasures. Commentationes Math. 18, 1975, 177-186. · Zbl 0339.28002
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