The symmetric Choquet integral with respect to Riesz-space-valued capacities

Antonio Boccuto; Beloslav Riečan

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 2, page 289-310
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Boccuto, Antonio, and Riečan, Beloslav. "The symmetric Choquet integral with respect to Riesz-space-valued capacities." Czechoslovak Mathematical Journal 58.2 (2008): 289-310. <http://eudml.org/doc/31211>.

@article{Boccuto2008,
abstract = {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.},
author = {Boccuto, Antonio, Riečan, Beloslav},
journal = {Czechoslovak Mathematical Journal},
keywords = {Riesz spaces; capacities; integration; symmetric Choquet integral; monotone and dominated convergence theorems; Riesz spaces; capacities; integration; symmetric Choquet integral; monotone and dominated convergence theorems},
language = {eng},
number = {2},
pages = {289-310},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The symmetric Choquet integral with respect to Riesz-space-valued capacities},
url = {http://eudml.org/doc/31211},
volume = {58},
year = {2008},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 2
SP - 289
EP - 310
AB - 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.
LA - eng
KW - Riesz spaces; capacities; integration; symmetric Choquet integral; monotone and dominated convergence theorems; Riesz spaces; capacities; integration; symmetric Choquet integral; monotone and dominated convergence theorems
UR - http://eudml.org/doc/31211
ER -

References

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  1. Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. 15 (1965), 599–631. (1965) MR0182661
  2. Riesz spaces, integration and sandwich theorems, Tatra Mountains Math. Publ. 3 (1993), 213–230. (1993) Zbl0815.28007MR1278536
  3. On the De Giorgi-Letta integral with respect to means with values in Riesz spaces, Real Analysis Exchange 2 (1995/6), 793–810. (1995/6) MR1407297
  4. 10.1007/BF02977030, Rend. Circ. Mat. Palermo, Ser. II 46 (1997), 255–278. (1997) MR1617345DOI10.1007/BF02977030
  5. The monotone integral with respect to Riesz space-valued capacities, Rend. Mat. (Roma) 16 (1996), 491–524. (1996) MR1422395
  6. On the De Giorgi-Letta integral in infinite dimensions, Atti Sem. Mat. Fis. Univ. Modena 4 (1992), 285–302. (1992) MR1179037
  7. 10.1090/S0002-9947-1957-0098165-6, Trans. Amer. Math. Soc. 86 (1957), 463–488. (1957) Zbl0087.04702MR0098165DOI10.1090/S0002-9947-1957-0098165-6
  8. Une notion générale de convergence faible pour des fonctions croissantes d’ensemble, Ann. Scuola Sup. Pisa 33 (1977), 61–99. (1977) 
  9. Non-Additive Measure and Integral, Kluwer, 1994. (1994) Zbl0826.28002MR1320048
  10. On the Choquet integral for Riesz space valued measure, Tatra Mountains Math. Publ. 19 (2000), 75–89. (2000) 
  11. 10.1007/BF01415993, ZOR Methods and Models of Operations Research 37 (1993), 231–256. (1993) MR1229945DOI10.1007/BF01415993
  12. 10.1007/BF01074898, Theory and Decision 34 (1993), 119–137. (1993) Zbl0780.90004MR1215033DOI10.1007/BF01074898
  13. Convex Cones, North-Holland Publ. Co., 1981. (1981) MR0640719
  14. 10.1007/BF02032160, Ann. Oper. Research 52 (1994), 43–65. (1994) MR1293559DOI10.1007/BF02032160
  15. Fuzzy Measures and Integrals: Theory and Applications, Studies in Fuzziness and Soft Computing, 40, Heidelberg, Physica-Verlag, 2000. (2000) MR1767776
  16. An Introduction to Stochastic Processes, North-Holland, New York, 1979. (1979) Zbl0418.60002MR0539142
  17. 10.1016/0165-0114(94)90116-3, Fuzzy Sets and Systems 62 (1994), 319–332. (1994) MR1276599DOI10.1016/0165-0114(94)90116-3
  18. 10.32917/hmj/1558306491, J. Sci. Hiroshima Univ. Ser. A 12 (1942), 17–35. (1942) MR0029087DOI10.32917/hmj/1558306491
  19. 10.1142/S0218488500000289, Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 8 (2000), 385–415. (2000) MR1781943DOI10.1142/S0218488500000289
  20. Null-additive Set Functions, Kluwer/Ister Science, 1995. (1995) Zbl0968.28010MR1368630
  21. 10.1016/0304-4068(92)90009-V, J. of Math. Econ. 21 (1992), 173–184. (1992) Zbl0761.90012MR1154830DOI10.1016/0304-4068(92)90009-V
  22. 10.1090/S0002-9939-1986-0835875-8, Proc. Am. Math. Soc. 97 (1986), 255–261. (1986) Zbl0687.28008MR0835875DOI10.1090/S0002-9939-1986-0835875-8
  23. Integral with respect to a pre-measure, Math. Slov. 29 (1979), 141–155. (1979) MR0578286
  24. Introduction to the Theory of Partially Ordered Spaces, Wolters-Noordhoff Sci. Publ., Groningen, 1967. (1967) Zbl0186.44601MR0224522

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