×

Convergence of series and submeasures of the set of positive integers. (English) Zbl 0755.40003

Let \(\mathbb{N}\) be the set of positive integers and \(P(\mathbb{N})\) the set of all subsets of \(\mathbb{N}\). A function \(m:P(\mathbb{N})\to[0,\infty)\) is a submeasure if \(A\subseteq B\Rightarrow m(A)\leq m(B)\) and \(m(A\cup B)\leq m(A)+m(B)\). The submeasure is called compact if \(m(\{a\})=0\) for every \(a\) in \(\mathbb{N}\), and for every \(\varepsilon>0\) there is a decomposition \(A_ 1\cup\cdots\cup A_ k=\mathbb{N}\) such that \(m(A_ i)<\varepsilon\) for each \(i\). The main theorem of the paper is that if \(m\) is a compact submeasure on \(P(\mathbb{N})\), then any infinite series of nonnegative elements \(\sum^ \infty_{n=1}a_ n\) converges if and only if \(m(A)=0\Rightarrow\sum_{n\in A}a_ n<\infty\). This extends a 1986 result of R. Estrada and R. P. Kanval, [Proc. Am. Math. Soc. 97, 682-686 (1986; Zbl 0592.40001)] as is shown by an example.

MSC:

40A05 Convergence and divergence of series and sequences
28A10 Real- or complex-valued set functions

Citations:

Zbl 0592.40001
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] ESTRADA R., KANVAL R. P.: Series that converge on sets of null density. Proc. of Amer. Math. Soc. 97, 1986, No 4, 682-680. · Zbl 0592.40001
[2] BUCK R. C.: The measure theoretic approach to density. Amer. J. Math. 68, 1946, 560-580. · Zbl 0061.07503
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.