| Title:
|
The symmetric Choquet integral with respect to Riesz-space-valued capacities (English) |
| Author:
|
Boccuto, Antonio |
| Author:
|
Riečan, Beloslav |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
289-310 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A definition of “Šipoš integral” is given, similarly to [3],[5],[10], for real-valued functions and with respect to Dedekind complete Riesz-space-valued “capacities”. A comparison of Choquet and Šipoš-type integrals is given, and some fundamental properties and some convergence theorems for the Šipoš integral are proved. (English) |
| Keyword:
|
Riesz spaces |
| Keyword:
|
capacities |
| Keyword:
|
integration |
| Keyword:
|
symmetric Choquet integral |
| Keyword:
|
monotone and dominated convergence theorems |
| MSC:
|
28A25 |
| MSC:
|
28A70 |
| MSC:
|
28B05 |
| MSC:
|
28C99 |
| MSC:
|
46G12 |
| idZBL:
|
Zbl 1174.28012 |
| idMR:
|
MR2411091 |
| . |
| Date available:
|
2009-09-24T11:54:55Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128259 |
| . |
| Reference:
|
[1] S. J. Bernau: Unique representation of Archimedean lattice groups and normal Archimedean lattice rings.Proc. London Math. Soc. 15 (1965), 599–631. MR 0182661 |
| Reference:
|
[2] A. Boccuto: Riesz spaces, integration and sandwich theorems.Tatra Mountains Math. Publ. 3 (1993), 213–230. Zbl 0815.28007, MR 1278536 |
| Reference:
|
[3] A. Boccuto and A. R. Sambucini: On the De Giorgi-Letta integral with respect to means with values in Riesz spaces.Real Analysis Exchange 2 (1995/6), 793–810. MR 1407297 |
| Reference:
|
[4] A. Boccuto and A. R. Sambucini: Comparison between different types of abstract integrals in Riesz spaces.Rend. Circ. Mat. Palermo, Ser. II 46 (1997), 255–278. MR 1617345, 10.1007/BF02977030 |
| Reference:
|
[5] A. Boccuto and A. R. Sambucini: The monotone integral with respect to Riesz space-valued capacities.Rend. Mat. (Roma) 16 (1996), 491–524. MR 1422395 |
| Reference:
|
[6] J. K. Brooks and A. Martellotti: On the De Giorgi-Letta integral in infinite dimensions.Atti Sem. Mat. Fis. Univ. Modena 4 (1992), 285–302. MR 1179037 |
| Reference:
|
[7] R. R. Christian: On order-preserving integration.Trans. Amer. Math. Soc. 86 (1957), 463–488. Zbl 0087.04702, MR 0098165, 10.1090/S0002-9947-1957-0098165-6 |
| Reference:
|
[8] E. De Giorgi and G. Letta: Une notion générale de convergence faible pour des fonctions croissantes d’ensemble.Ann. Scuola Sup. Pisa 33 (1977), 61–99. |
| Reference:
|
[9] D. Denneberg: Non-Additive Measure and Integral.Kluwer, 1994. Zbl 0826.28002, MR 1320048 |
| Reference:
|
[10] M. Duchoň, J. Haluška and B. Riečan: On the Choquet integral for Riesz space valued measure.Tatra Mountains Math. Publ. 19 (2000), 75–89. |
| Reference:
|
[11] R. Dyckerhoff and K. Mosler: Stochastic dominance with nonadditive probabilities.ZOR Methods and Models of Operations Research 37 (1993), 231–256. MR 1229945, 10.1007/BF01415993 |
| Reference:
|
[12] P. C. Fishburn: The axioms and algebra of ambiguity.Theory and Decision 34 (1993), 119–137. Zbl 0780.90004, MR 1215033, 10.1007/BF01074898 |
| Reference:
|
[13] B. Fuchssteiner and W. Lusky: Convex Cones.North-Holland Publ. Co., 1981. MR 0640719 |
| Reference:
|
[14] I. Gilboa and D. Schmeidler: Additive representation of non-additive measures and the Choquet integral.Ann. Oper. Research 52 (1994), 43–65. MR 1293559, 10.1007/BF02032160 |
| Reference:
|
[15] M. Grabisch, T. Murofushi and M. Sugeno (Eds.): Fuzzy Measures and Integrals: Theory and Applications.Studies in Fuzziness and Soft Computing, 40, Heidelberg, Physica-Verlag, 2000. MR 1767776 |
| Reference:
|
[16] D. Kannan: An Introduction to Stochastic Processes.North-Holland, New York, 1979. Zbl 0418.60002, MR 0539142 |
| Reference:
|
[17] X. Liu and G. Zhang: Lattice-valued fuzzy measure and lattice-valued fuzzy integral.Fuzzy Sets and Systems 62 (1994), 319–332. MR 1276599, 10.1016/0165-0114(94)90116-3 |
| Reference:
|
[18] F. Maeda and T. Ogasawara: Representation of vector lattices.J. Sci. Hiroshima Univ. Ser. A 12 (1942), 17–35. MR 0029087, 10.32917/hmj/1558306491 |
| Reference:
|
[19] T. Murofushi and M. Sugeno: Choquet integral models and independence concepts in multiattribute utility theory.Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 8 (2000), 385–415. MR 1781943, 10.1142/S0218488500000289 |
| Reference:
|
[20] E. Pap: Null-additive Set Functions.Kluwer/Ister Science, 1995. Zbl 0968.28010, MR 1368630 |
| Reference:
|
[21] M. Scarsini: Dominance conditions in non-additive expected utility theory.J. of Math. Econ. 21 (1992), 173–184. Zbl 0761.90012, MR 1154830, 10.1016/0304-4068(92)90009-V |
| Reference:
|
[22] D. Schmeidler: Integral representation without additivity.Proc. Am. Math. Soc. 97 (1986), 255–261. Zbl 0687.28008, MR 0835875, 10.1090/S0002-9939-1986-0835875-8 |
| Reference:
|
[23] J. Šipoš: Integral with respect to a pre-measure.Math. Slov. 29 (1979), 141–155. MR 0578286 |
| Reference:
|
[24] B. Z. Vulikh: Introduction to the Theory of Partially Ordered Spaces.Wolters-Noordhoff Sci. Publ., Groningen, 1967. Zbl 0186.44601, MR 0224522 |
| . |
