Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 1085-1100, 2012

Decomposition of $\ell$-group-valued measures

Giuseppina Barbieri, Antonietta Valente, Hans Weber

Giuseppina Barbieri, University of Udine, Via delle Scienze 206, I-33100 Udine, Italy, e-mail: giuseppina.barbieri@uniud.it; Antonietta Valente, Via Vespucci Amerigo 13, I-85100 Potenza, Italy; Hans Weber, University of Udine, Via delle Scienze 206, I-33100 Udine, Italy, e-mail: hans.weber@uniud.it

Abstract: We deal with decomposition theorems for modular measures $\mu L\rightarrow G$ 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.

Keywords: D-lattice, measure, lattice ordered group, decomposition, Hammer-Sobczyk decomposition

Classification (MSC 2010): 28B15, 06C15


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