Decomposition of $\ell$-group-valued measures

Barbieri, Giuseppina, Valente, Antonietta, and Weber, Hans. "Decomposition of $\ell $-group-valued measures." Czechoslovak Mathematical Journal 62.4 (2012): 1085-1100. <http://eudml.org/doc/246601>.

@article{Barbieri2012,
abstract = {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.},
author = {Barbieri, Giuseppina, Valente, Antonietta, Weber, Hans},
journal = {Czechoslovak Mathematical Journal},
keywords = {D-lattice; measure; lattice ordered group; decomposition; Hammer-Sobczyk decomposition; D-lattice; measure; lattice ordered group; Hammer-Sobczyk decomposition},
language = {eng},
number = {4},
pages = {1085-1100},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Decomposition of $\ell $-group-valued measures},
url = {http://eudml.org/doc/246601},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Barbieri, Giuseppina
AU - Valente, Antonietta
AU - Weber, Hans
TI - Decomposition of $\ell $-group-valued measures
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 4
SP - 1085
EP - 1100
AB - 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.
LA - eng
KW - D-lattice; measure; lattice ordered group; decomposition; Hammer-Sobczyk decomposition; D-lattice; measure; lattice ordered group; Hammer-Sobczyk decomposition
UR - http://eudml.org/doc/246601
ER -