Add to ORKGAbstract. We deal with decomposition theorems for modular measures µ: L → G de-fined on a D-lattice with values in a Dedekind complete ℓ-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete ℓ-groups, several decomposition the-orems 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 ℓ-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...