# Article

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Keywords:
D-lattice; measure; lattice ordered group; decomposition; Hammer-Sobczyk decomposition
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.
References:
[1] Avallone, A., Vitolo, P.: Congruences and ideals of effect algebras. Order 20 (2003), 67-77. DOI 10.1023/A:1024458125510 | MR 1993411 | Zbl 1030.03047
[2] Avallone, A., Barbieri, G., Vitolo, P.: On the Alexandroff decomposition theorem. Math. Slovaca 58 (2008), 185-200. DOI 10.2478/s12175-008-0067-2 | MR 2391213 | Zbl 1174.28010
[3] Avallone, A., Barbieri, G., Vitolo, P., Weber, H.: Decomposition of effect algebras and the Hammer-Sobczyk theorem. Algebra Univers. 60 (2009), 1-18. MR 2480629 | Zbl 1171.28004
[4] Birkhoff, G.: Lattice Theory. American Mathematical Society New York (1940). MR 0001959 | Zbl 0063.00402
[5] Rao, K. P. S. Bhaskara, Rao, M. Bhaskara: Theory of Charges. A Study of Finitely Additive Measures. Pure and Applied Mathematics, 109 Academic Press, a Subsidiary of Harcourt Brace Jovanovich, Publishers (1983). MR 0751777
[6] Boccuto, A., Candeloro, D.: Sobczyk-Hammer decompositions and convergence theorems for measures with values in $\ell$-groups. Real Anal. Exch. 33 (2008), 91-106. MR 2402865
[7] Iglesias, M. Congost: Measures and probabilities in ordered structures. Stochastica 5 (1981), 45-68. MR 0625841
[8] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Dordrecht: Kluwer Academic Publishers Bratislava: Ister Science (2000). MR 1861369
[9] Fleischer, I., Traynor, T.: Group-valued modular functions. Algebra Univers. 14 (1982), 287-291. DOI 10.1007/BF02483932 | MR 0654397 | Zbl 0458.06004
[10] Glass, A. M. W., Holland, W. C.: Lattice-ordered Groups. Advances and Techniques Kluwer Academic Publishers (1989). MR 1036072 | Zbl 0705.06001
[11] Hammer, P. C., Sobczyk, A.: The ranges of additive set functions. Duke Math. J. 11 (1944), 847-851. DOI 10.1215/S0012-7094-44-01173-7 | MR 0011165 | Zbl 0061.09803
[12] Riesz, F.: Sur quelques notions fondamentales dans la théorie générale des opérations linéaires. Ann. Math. 41 (1940), 174-206 French. DOI 10.2307/1968825 | MR 0000902 | Zbl 0022.31802
[13] Schmidt, K. D.: Jordan Decompositions of Generalized Vector Measures. Pitman Research Notes in Mathematics Series, 214 Harlow: Longman Scientific & Technical; New York etc.: John Wiley & Sons, Inc. (1989). MR 1028550 | Zbl 0692.28004
[14] Schmidt, K. D.: Decomposition and extension of abstract measures in Riesz spaces. Rend. Ist. Mat. Univ. Trieste 29 (1998), 135-213. MR 1696025 | Zbl 0929.28009

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