Decomposition of a%26quot;%26quot;-group-valued measures

作者：Barbieri Giuseppina^*; Valente Antonietta; Weber Hans

来源：Czechoslovak Mathematical Journal, 2012, 62(4): 1085-1100.

DOI：10.1007/s10587-012-0065-y

• D-lattice
• measure
• lattice ordered group
• decomposition
• Hammer-Sobczyk decomposition

We deal with decomposition theorems for modular measures A mu: L -%26gt; G defined on a D-lattice with values in a Dedekind complete a%26quot;%26quot;-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete a%26quot;%26quot;-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%26quot;%26quot;-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.

• 出版日期2012-12