Geodesic
The symmetric Choquet integral with respect to Riesz-space-valued capacities
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 289-310 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 28A25, 28A70, 28B05, 28C99, 46G12
Keywords: Riesz spaces; capacities; integration; symmetric Choquet integral; monotone and dominated convergence theorems 
@article{CMJ_2008_58_2_a0,
     author = {Boccuto, Antonio and Rie\v{c}an, Beloslav},
     title = {The symmetric {Choquet} integral with respect to {Riesz-space-valued} capacities},
     journal = {Czechoslovak Mathematical Journal},
     pages = {289--310},
     year = {2008},
     volume = {58},
     number = {2},
     mrnumber = {2411091},
     zbl = {1174.28012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a0/}
}
TY  - JOUR
AU  - Boccuto, Antonio
AU  - Riečan, Beloslav
TI  - The symmetric Choquet integral with respect to Riesz-space-valued capacities
JO  - Czechoslovak Mathematical Journal
PY  - 2008
SP  - 289
EP  - 310
VL  - 58
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a0/
LA  - en
ID  - CMJ_2008_58_2_a0
ER  - 
%0 Journal Article
%A Boccuto, Antonio
%A Riečan, Beloslav
%T The symmetric Choquet integral with respect to Riesz-space-valued capacities
%J Czechoslovak Mathematical Journal
%D 2008
%P 289-310
%V 58
%N 2
%U http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a0/
%G en
%F CMJ_2008_58_2_a0
Boccuto, Antonio; Riečan, Beloslav. The symmetric Choquet integral with respect to Riesz-space-valued capacities. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 289-310. http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a0/

[1] S. J. Bernau: Unique representation of Archimedean lattice groups and normal Archimedean lattice rings. Proc. London Math. Soc. 15 (1965), 599–631. | MR

[2] A. Boccuto: Riesz spaces, integration and sandwich theorems. Tatra Mountains Math. Publ. 3 (1993), 213–230. | MR | Zbl

[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

[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. | DOI | MR

[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

[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

[7] R. R. Christian: On order-preserving integration. Trans. Amer. Math. Soc. 86 (1957), 463–488. | DOI | MR | Zbl

[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.

[9] D. Denneberg: Non-Additive Measure and Integral. Kluwer, 1994. | MR | Zbl

[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.

[11] R. Dyckerhoff and K. Mosler: Stochastic dominance with nonadditive probabilities. ZOR Methods and Models of Operations Research 37 (1993), 231–256. | DOI | MR

[12] P. C. Fishburn: The axioms and algebra of ambiguity. Theory and Decision 34 (1993), 119–137. | DOI | MR | Zbl

[13] B. Fuchssteiner and W. Lusky: Convex Cones. North-Holland Publ. Co., 1981. | MR

[14] I. Gilboa and D. Schmeidler: Additive representation of non-additive measures and the Choquet integral. Ann. Oper. Research 52 (1994), 43–65. | DOI | MR

[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

[16] D. Kannan: An Introduction to Stochastic Processes. North-Holland, New York, 1979. | MR | Zbl

[17] X. Liu and G. Zhang: Lattice-valued fuzzy measure and lattice-valued fuzzy integral. Fuzzy Sets and Systems 62 (1994), 319–332. | DOI | MR

[18] F. Maeda and T. Ogasawara: Representation of vector lattices. J. Sci. Hiroshima Univ. Ser. A 12 (1942), 17–35. | DOI | MR

[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. | DOI | MR

[20] E. Pap: Null-additive Set Functions. Kluwer/Ister Science, 1995. | MR | Zbl

[21] M. Scarsini: Dominance conditions in non-additive expected utility theory. J. of Math. Econ. 21 (1992), 173–184. | DOI | MR | Zbl

[22] D. Schmeidler: Integral representation without additivity. Proc. Am. Math. Soc. 97 (1986), 255–261. | DOI | MR | Zbl

[23] J. Šipoš: Integral with respect to a pre-measure. Math. Slov. 29 (1979), 141–155. | MR

[24] B. Z. Vulikh: Introduction to the Theory of Partially Ordered Spaces. Wolters-Noordhoff Sci. Publ., Groningen, 1967. | MR | Zbl