zbMATH — the first resource for mathematics

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