Geodesic
An extension theorem for modular measures on effect algebras
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 707-719 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We prove an extension theorem for modular measures on lattice ordered effect algebras. This is used to obtain a representation of these measures by the classical ones. With the aid of this theorem we transfer control theorems, Vitali-Hahn-Saks, Nikodým theorems and range theorems to this setting.
We prove an extension theorem for modular measures on lattice ordered effect algebras. This is used to obtain a representation of these measures by the classical ones. With the aid of this theorem we transfer control theorems, Vitali-Hahn-Saks, Nikodým theorems and range theorems to this setting.
Classification : 06C15, 28E99
Keywords: effect algebras; modular measures; extension; Vitali-Hahn-Saks theorem; Nikodým theorem; decomposition theorem; control theorems; range; Liapunoff theorem 
@article{CMJ_2009_59_3_a11,
     author = {Barbieri, Giuseppina},
     title = {An extension theorem for modular measures on effect algebras},
     journal = {Czechoslovak Mathematical Journal},
     pages = {707--719},
     year = {2009},
     volume = {59},
     number = {3},
     mrnumber = {2545651},
     zbl = {1224.28037},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2009_59_3_a11/}
}
TY  - JOUR
AU  - Barbieri, Giuseppina
TI  - An extension theorem for modular measures on effect algebras
JO  - Czechoslovak Mathematical Journal
PY  - 2009
SP  - 707
EP  - 719
VL  - 59
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2009_59_3_a11/
LA  - en
ID  - CMJ_2009_59_3_a11
ER  - 
%0 Journal Article
%A Barbieri, Giuseppina
%T An extension theorem for modular measures on effect algebras
%J Czechoslovak Mathematical Journal
%D 2009
%P 707-719
%V 59
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2009_59_3_a11/
%G en
%F CMJ_2009_59_3_a11
Barbieri, Giuseppina. An extension theorem for modular measures on effect algebras. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 707-719. http://geodesic.mathdoc.fr/item/CMJ_2009_59_3_a11/

[1] Avallone, A.: Lattice uniformities on orthomodular structures. Math. Slovaca 51 (2001), 403-419. | MR | Zbl

[2] Avallone, A.: Cafiero and Nikodým boundedness theorems in effect algebras. Ital. J. Pure Appl. Math. 20 (2006), 203-214. | MR | Zbl

[3] Avallone, A.: Separating points of measures on effect algebras. Math. Slovaca 57 (2007), 129-140. | DOI | MR | Zbl

[4] Avallone, A., Barbieri, G., Vitolo, P.: Hahn decomposition of modular measures and applications. Comment. Math. Prace Mat. 43 (2003), 149-168. | MR | Zbl

[5] Avallone, A., Barbieri, G., Vitolo, P., Weber, H.: Decomposition of effect algebras and the Hammer-Sobczyk theorem. (to appear) in Algebra Universalis. | MR | Zbl

[6] Avallone, A., Basile, A.: On a Marinacci uniqueness theorem for measures. J. Math. Anal. Appl. 286 (2003), 378-390. | DOI | MR | Zbl

[7] Avallone, A., Simone, A. De, Vitolo, P.: Effect algebras and extensions of measures. Bollettino U.M.I. 9-B (2006), 423-444. | MR | Zbl

[8] Avallone, A., Rinauro, S., Vitolo, P.: Boundedness and convergence theorems in effect algebras. Tatra Mountains Math. Publ. 37 (2007), 1-16. | MR | Zbl

[9] Avallone, A., Vitolo, P.: Decomposition and control theorems for measures on effect algebras. Sci. Math. Japon 58 (2003), 1-14. | MR

[10] Avallone, A., Vitolo, P.: Congruences and ideals of effect algebras. Order 20 (2003), 67-77. | MR | Zbl

[11] Basile, A.: Controls of families of finitely additive functions. Ricerche Mat. 35 (1986), 291-302. | MR | Zbl

[12] Barbieri, G.: A note on fuzzy measures. J. Electr. Eng. 52 (2001), 67-70. | MR | Zbl

[13] Barbieri, G.: Lyapunov's theorem for measures on $D$-posets. Internat. J. Theoret. Phys. 43 (2004), 1613-1623. | MR | Zbl

[14] Rao, K. P. S. Bhaskara, Rao, M. Bhaskara: Theory of Charges. A Study of Finitely Additive Measures. Academic Press, Inc. New York (1983). | MR

[15] Bennett, M. K., Foulis, D. J.: Effect algebras and unsharp quantum logics. Special issue dedicated to Constantin Piron on the occasion of his sixtieth birthday. Found. Phys. 24 (1994), 1331-1352. | MR

[16] Diestel, J., Uhl, J. J.: Vector Measures. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I. (1977). | MR | Zbl

[17] Dvurecenskij, A., Pulmannovà, S.: New Trends in Quantum Structures. Kluwer Academic Publishers, Bratislava (2000). | MR

[18] Fischer, W., Schoeler, U.: The range of vector measures in Orlicz spaces. Studia Math. 59 (1976), 53-61. | MR

[19] Fleischer, I., Traynor, T.: Equivalence of group-valued measures on an abstract lattice. Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 549-556. | MR | Zbl

[20] Foulis, D., Greechie, R. J., Pulmannová, S.: The center of an effect algebra. Order 12 (1995), 91-106. | MR

[21] Kadets, V. M.: A remark on Lyapunov's theorem on a vector measure. Russian Funktsional. Anal. i Prilozhen. 25 (1991), 78-80 Translation in Funct. Anal. Appl. 25 (1992), 295-297. | MR

[22] Kadets, V. M., Shekhtman, G.: Lyapunov's theorem for $l\sb p$-valued measures. Russian Algebra i Analiz 4 (1992), 148-154 Translation in St. Petersburg Math. J. 4 (1993), 961-966. | MR

[23] Kluvánek, I.: The range of a vector-valued measure. Math. Systems Theory 7 (1973), 44-54. | MR

[24] Shapiro, J.: On convexity and compactness in $F$-spaces with bases. Indiana Univ. Math. J. 21 (1971/72), 1073-1090. | MR

[25] Uhl, J.: The range of a vector-valued measure. Proc. Amer. Math. Soc. 23 (1969), 158-163. | MR | Zbl

[26] Weber, H.: Group- and vector-valued $s$-bounded contents. Lecture Notes in Math. 1089, Springer (1984), 181-198. | MR | Zbl

[27] Weber, H.: Compactness in spaces of group-valued contents, the Vitali-Hahn-Saks theorem and Nikodým's boundedness theorem. Rocky Mountain J. Math. 16 (1986), 253-275. | MR | Zbl

[28] Weber, H.: Uniform lattices. I. A generalization of topological Riesz spaces and topological Boolean rings. Ann. Mat. Pura Appl. 160 (1991), 1992 547-570. | MR | Zbl

[29] Weber, H.: Uniform lattices. II. Order continuity and exhaustivity. Ann. Mat. Pura Appl. 165 (1993), 133-158. | MR | Zbl

[30] Weber, H.: On modular functions. Funct. Approx. Comment. Math. 24 (1996), 35-52. | MR | Zbl

[31] Weber, H.: Uniform lattices and modular functions. Atti Sem. Mat. Fis. Univ. Modena XLVII (1999), 159-182. | MR | Zbl

[32] Weber, H.: FN-topologies and group-valued measures. Handbook of measure theory, Vol. I, II. 703-743 North-Holland, Amsterdam (2002). | MR

[33] Weber, H.: Two extension theorems. Modular functions on complemented lattices. Czech. Math. J. 52 (2002), 55-74. | MR | Zbl