| Title:
|
Two extension theorems. Modular functions on complemented lattices (English) |
| Author:
|
Weber, Hans |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
52 |
| Issue:
|
1 |
| Year:
|
2002 |
| Pages:
|
55-74 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We prove an extension theorem for modular functions on arbitrary lattices and an extension theorem for measures on orthomodular lattices. The first is used to obtain a representation of modular vector-valued functions defined on complemented lattices by measures on Boolean algebras. With the aid of this representation theorem we transfer control measure theorems, Vitali-Hahn-Saks and Nikodým theorems and the Liapunoff theorem about the range of measures to the setting of modular functions on complemented lattices. (English) |
| Keyword:
|
complemented lattices |
| Keyword:
|
orthomodular lattices |
| Keyword:
|
exhaustive modular functions |
| Keyword:
|
measures |
| Keyword:
|
extension |
| Keyword:
|
Vitali-Hahn-Saks theorem |
| Keyword:
|
Nikodým theorems |
| Keyword:
|
Liapunoff theorem |
| MSC:
|
06B30 |
| MSC:
|
06C15 |
| MSC:
|
28E99 |
| idZBL:
|
Zbl 0998.06006 |
| idMR:
|
MR1885457 |
| . |
| Date available:
|
2009-09-24T10:49:07Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127702 |
| . |
| Reference:
|
[1] A. Avallone: Liapunov theorem for modular functions.Internat. J. Theoret. Phys. 34 (1995), 1197–1204. Zbl 0841.28007, MR 1353662 |
| Reference:
|
[2] A. Avallone: Nonatomic vector valued modular functions.Mathematicae Polonae, Ser. I: Comment. Math. 39 (1999), 23–36. Zbl 0987.28012, MR 1739014 |
| Reference:
|
[3] A. Avallone, G. Barbieri and R. Cilia: Control and separating points of modular functions.Math. Slovaca 49 (1999), 155–182. MR 1696950 |
| Reference:
|
[4] A. Avallone and J. Hamhalter: Extension theorems (Vector measures on quantum logics).Czechoslovak Math. J. 46 (1996), 179–192. MR 1371699 |
| Reference:
|
[5] A. Avallone and M. A. Lepellere: Modular functions: Uniform boundedness and compactness.Rend. Circ. Mat. Palermo 47 (1998), 221–264. MR 1633479, 10.1007/BF02844366 |
| Reference:
|
[6] A. Basile: Controls of families of finitely additive functions.Ricerche Mat. 35 (1986), 291–302. Zbl 0648.28007, MR 0932439 |
| Reference:
|
[7] G. Birkhoff: Lattice Theory.AMS colloquium Publications, Providence, Rhode Island, 25 (1984). MR 0751233 |
| Reference:
|
[8] J. K. Brooks, D. Candeloro and A. Martellotti: On finitely additive measures in nuclear spaces.Atti Sem. Mat. Fis. Univ. Modena 46 (1998), 37–50. MR 1628585 |
| Reference:
|
[9] I. Fleischer and T. Traynor: Equivalence of group-valued measures on an abstract lattice.Bull. Acad. Pol. Sci., Sér. Sci. Math. 28 (1980), 549–556. MR 0628641 |
| Reference:
|
[10] I. Fleischer and T. Traynor: Group-valued modular functions.Algebra Universalis 14 (1982), 287–291. MR 0654397, 10.1007/BF02483932 |
| Reference:
|
[11] G. Grätzer: General Lattice Theory. Pure and Applied Mathematical Series.Academic Press, San Diego, 1978. MR 0509213 |
| Reference:
|
[12] V. M. Kadets: A remark on Lyapunov theorem on vector measures.Funct. Anal. Appl. 25 (1991), 295–297. MR 1167727 |
| Reference:
|
[13] V. M. Kadets and G. Shekhtman: The Lyapunov theorem for $\ell _p$-valued measures.St Petersburg Math. J. 4 (1993), 961–966. MR 1202728 |
| Reference:
|
[14] J. J. Uhl: The range of vector-valued measures.Proc. Amer. Math. Soc. 23 (1969), 158–163. MR 0264029, 10.1090/S0002-9939-1969-0264029-1 |
| Reference:
|
[15] T. Traynor: The Lebesgue decomposition for group-valued set functions.Trans. Amer. Math. Soc. 220 (1976), 307–319. Zbl 0334.28010, MR 0419725, 10.1090/S0002-9947-1976-0419725-8 |
| Reference:
|
[16] H. Weber: Group- and vector-valued $s$-bounded contents.Measure Theory, Proc. Conf. (Oberwolfach 1983). Zbl 0552.28011, MR 0786697 |
| Reference:
|
[17] H. Weber: Uniform lattices I: A generalization of topological Riesz spaces and topological Boolean rings; Uniform lattices II: Order continuity and exhaustivity.Ann. Mat. Pura Appl. 160 (1991 1991), 347–370. MR 1163215, 10.1007/BF01764134 |
| Reference:
|
[18] H. Weber: Valuations on complemented lattices.Inter. J. Theoret. Phys. 34 (1995), 1799–1806. Zbl 0843.06005, MR 1353726 |
| Reference:
|
[19] H. Weber: On modular functions.Funct. Approx. Comment. Math. 24 (1996), 35–52. Zbl 0887.06011, MR 1453447 |
| Reference:
|
[20] H. Weber: Lattice uniformities and modular functions.Atti Sem. Mat. Fis. Univ. Modena XLVII (1999), 159–182. MR 1694416 |
| Reference:
|
[21] H. Weber: Complemented uniform lattices.Topology Appl. 105 (2000), 47–64. Zbl 1121.54312, MR 1761086, 10.1016/S0166-8641(99)00049-8 |
| . |
