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On the lattice group valued submeasures. (English) Zbl 0760.28008
Let $$G$$ be a weakly $$\sigma$$-distributive Dedekind complete l-group, let $$X$$ be a set of cardinality $$2^{\aleph_ 0}$$, and let $$m: 2^ X\to G$$ be a submeasure, i.e., $$m$$ is nonnegative, increasing, subadditive and $$\bigwedge^ \infty_{n=1} m(A_ n)=0$$ whenever $$X\supset A_ 1\supset A_ 2\supset\cdots$$ and $$\bigcap^ \infty_{n=1} A_ n=\emptyset$$. Generalizing slightly a result of Z. Riečanová [Math. Slovaca 39, No. 1, 91-97 (1989; Zbl 0676.28005)], the author proves that, under the continuum hypothesis, $$m(X)=0$$ provided that $$m(\{x\})=0$$ for all $$x\in X$$. For $$G=\mathbb{R}$$ this is the classical Banach-Kuratowski theorem. Both Riečanová and the author base their proofs on a combinatorial lemma due to Banach and Kuratowski, but author’s proof is more direct.

##### MSC:
 28B15 Set functions, measures and integrals with values in ordered spaces 28B10 Group- or semigroup-valued set functions, measures and integrals 28A12 Contents, measures, outer measures, capacities 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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