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Notes on lattice-valued measures. (English) Zbl 0568.28010
Let \({\mathcal R}\) be a ring of subsets of some set \(\Omega\) and let G be an order complete lattice-ordered group. The author proves that each positive, order countably additive set function \({\mathcal R}\to G\) has a unique extension to the \(\sigma\)-algebra generated by \({\mathcal R}\), provided that either G is weakly (\(\sigma\),\(\infty)\)-distributive or G has the countable-sup property and the family of all positive order continuous linear functionals on G is separating. The results of this paper are related to the author’s earlier work [Math. Slovaca 33, 153-163 (1983; Zbl 0519.28004)] where the extension problem was studied in a more general context. The proofs are purely algebraic and they thus differ from those of T. V. Panchapagesan and Shivappa Veerappa Palled [Math. Slovaca 33, 269-292 (1983; Zbl 0528.28008)] who considered the case where G is a vector lattice and used the Krein-Kakutani representation theorem.
Reviewer: K.D.Schmidt

28B15 Set functions, measures and integrals with values in ordered spaces
28B10 Group- or semigroup-valued set functions, measures and integrals