## 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

### Keywords:

compact submeasures; infinite series

Zbl 0592.40001
Full Text:

### 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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