| Title:
|
Decomposition of $\ell $-group-valued measures (English) |
| Author:
|
Barbieri, Giuseppina |
| Author:
|
Valente, Antonietta |
| Author:
|
Weber, Hans |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
4 |
| Year:
|
2012 |
| Pages:
|
1085-1100 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
D-lattice |
| Keyword:
|
measure |
| Keyword:
|
lattice ordered group |
| Keyword:
|
decomposition |
| Keyword:
|
Hammer-Sobczyk decomposition |
| MSC:
|
06C15 |
| MSC:
|
06F15 |
| MSC:
|
28B10 |
| MSC:
|
28B15 |
| idZBL:
|
Zbl 1274.28025 |
| idMR:
|
MR3010258 |
| DOI:
|
10.1007/s10587-012-0065-y |
| . |
| Date available:
|
2012-11-10T21:46:39Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143046 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| . |
