Decomposition of a""-group-valued measures

We deal with decomposition theorems for modular measures A mu: L -> G defined on a D-lattice with values in a Dedekind complete a""-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete a""-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result-also based on the band decomposition theorem of Riesz-is the Hammer-Sobczyk decomposition for a""-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.