The Choquet integral in Riesz space

A comprehensive discussion of the theory of Choquet integration in a Riesz space is given. In particular, it is proved that the monotone convergence theorem, the Fatou lemma, and the dominated convergence theorem are still valid for Riesz space-valued non-additive measures if we assume that the Riesz space has a new property concerning the cardinality of the set of points of discontinuity of a monotone function.