This paper is concerned with lattice-group valued measures for wich the σ-additivity is defined by means of the order convergence properties. In the first section we treat the analogues for such order-measures with values in a Dedekind complete lattice-group of the Jordan, Lebesgue and Yosida-Hewitt descompositions. The second section deals with the construction of an integral for functions with respect to an order-measure, both taking their values in a Dedekind σ-complete lattice-ring. Analogues of the monotone-convergence, dominated-convergence and Fatou theorems are obtained.