Abstract
We deal with decomposition theorems for modular measures µ: L → G defined 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 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 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 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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Barbieri, G., Valente, A. & Weber, H. Decomposition of ℓ-group-valued measures. Czech Math J 62, 1085–1100 (2012). https://doi.org/10.1007/s10587-012-0065-y
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DOI: https://doi.org/10.1007/s10587-012-0065-y