Decomposition of $\ell $-group-valued measures

-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.

MR Zbl

DOI : 10.1007/s10587-012-0065-y

Classification : 06C15, 06F15, 28B10, 28B15
Mots-clés : D-lattice; measure; lattice ordered group; decomposition; Hammer-Sobczyk decomposition

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     author = {Barbieri, Giuseppina and Valente, Antonietta and Weber, Hans},
     title = {Decomposition of $\ell $-group-valued measures},
     journal = {Czechoslovak Mathematical Journal},
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     zbl = {1274.28025},
     language = {en},
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Barbieri, Giuseppina; Valente, Antonietta; Weber, Hans. Decomposition of $\ell $-group-valued measures. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 1085-1100. doi : 10.1007/s10587-012-0065-y. https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-012-0065-y/

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