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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
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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
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