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D-lattice; measure; lattice ordered group; decomposition; Hammer-Sobczyk decomposition
We deal with decomposition theorems for modular measures $\mu \colon L\rightarrow G$ defined on a D-lattice with values in a Dedekind complete $\ell $-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete $\ell $-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 $\ell $-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.
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