The Choquet integral as Lebesgue integral and related inequalities

Radko Mesiar; Jun Li; Endre Pap

Kybernetika (2010)

  • Volume: 46, Issue: 6, page 1098-1107
  • ISSN: 0023-5954

Abstract

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The integral inequalities known for the Lebesgue integral are discussed in the framework of the Choquet integral. While the Jensen inequality was known to be valid for the Choquet integral without any additional constraints, this is not more true for the Cauchy, Minkowski, Hölder and other inequalities. For a fixed monotone measure, constraints on the involved functions sufficient to guarantee the validity of the discussed inequalities are given. Moreover, the comonotonicity of the considered functions is shown to be a sufficient constraint ensuring the validity of all discussed inequalities for the Choquet integral, independently of the underlying monotone measure.

How to cite

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Mesiar, Radko, Li, Jun, and Pap, Endre. "The Choquet integral as Lebesgue integral and related inequalities." Kybernetika 46.6 (2010): 1098-1107. <http://eudml.org/doc/196461>.

@article{Mesiar2010,
abstract = {The integral inequalities known for the Lebesgue integral are discussed in the framework of the Choquet integral. While the Jensen inequality was known to be valid for the Choquet integral without any additional constraints, this is not more true for the Cauchy, Minkowski, Hölder and other inequalities. For a fixed monotone measure, constraints on the involved functions sufficient to guarantee the validity of the discussed inequalities are given. Moreover, the comonotonicity of the considered functions is shown to be a sufficient constraint ensuring the validity of all discussed inequalities for the Choquet integral, independently of the underlying monotone measure.},
author = {Mesiar, Radko, Li, Jun, Pap, Endre},
journal = {Kybernetika},
keywords = {Choquet integral; comonotone functions; integral inequalities; monotone measure; modularity; Choquet integral; comonotone functions; integral inequalities; monotone measure; modularity},
language = {eng},
number = {6},
pages = {1098-1107},
publisher = {Institute of Information Theory and Automation AS CR},
title = {The Choquet integral as Lebesgue integral and related inequalities},
url = {http://eudml.org/doc/196461},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Mesiar, Radko
AU - Li, Jun
AU - Pap, Endre
TI - The Choquet integral as Lebesgue integral and related inequalities
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 6
SP - 1098
EP - 1107
AB - The integral inequalities known for the Lebesgue integral are discussed in the framework of the Choquet integral. While the Jensen inequality was known to be valid for the Choquet integral without any additional constraints, this is not more true for the Cauchy, Minkowski, Hölder and other inequalities. For a fixed monotone measure, constraints on the involved functions sufficient to guarantee the validity of the discussed inequalities are given. Moreover, the comonotonicity of the considered functions is shown to be a sufficient constraint ensuring the validity of all discussed inequalities for the Choquet integral, independently of the underlying monotone measure.
LA - eng
KW - Choquet integral; comonotone functions; integral inequalities; monotone measure; modularity; Choquet integral; comonotone functions; integral inequalities; monotone measure; modularity
UR - http://eudml.org/doc/196461
ER -

References

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  11. Roman-Flores, H., Flores–Franuli, A., Chalco-Cano, Y., 10.1016/j.ins.2007.02.006, Inform. Sci. 177 (2007), 3192–3201. (2007) MR2340853DOI10.1016/j.ins.2007.02.006
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  13. Schmeidler, D., 10.2307/1911053, Econometrica 57 (1989), 571–587. (1989) Zbl0672.90011MR0999273DOI10.2307/1911053
  14. Sugeno, M., Narukawa, Y., Murofushi, T., Choquet integral and fuzzy measures on locally compact space, Fuzzy Sets and Systems 99, (1998), 2, 205–211. (1998) Zbl0977.28012MR1646177
  15. Wang, R.-S., Some inequalities and convergence theorems for Choquet integral, J. Appl. Math. Comput., DOI 10.1007/212190/009/0358-y. (1007) 
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