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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

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