Measure and integration in Riesz spaces



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Let [Omega] be a set, A an algebra of subsets of [Omega] , and V a complete Riesz space. The set of bounded, finitely additive set functions from A into V , denoted by Mlowered F , is a complete Riesz space. The set Mlowered C consisting of the countably additive members of Mlowered F is a band in Mlowered F . Suppose S is a sigma algebra of subsets of [Omega] and [mu] is a positive member of Mlowered C . A V-valued integral with respect to [mu] is defined on a subspace of the real valued functions on [Omega] . Properties of the integral are investigated and are related to the structure of Mlowered C . Positive, linear, sequentially order continuous functions from the ideal of bounded elements of a Riesz space with the principal projection property and a weak unit into a complete Riesz space are characterized in terms of an integral.