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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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References:
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