Geodesic
The Choquet integral as Lebesgue integral and related inequalities
Kybernetika, Tome 46 (2010) no. 6, pp. 1098-1107

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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.
Classification : 26D15, 28E10
Keywords: Choquet integral; comonotone functions; integral inequalities; monotone measure; modularity 
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     author = {Mesiar, Radko and Li, Jun and Pap, Endre},
     title = {The {Choquet} integral as {Lebesgue} integral and related inequalities},
     journal = {Kybernetika},
     pages = {1098--1107},
     publisher = {mathdoc},
     volume = {46},
     number = {6},
     year = {2010},
     mrnumber = {2797430},
     zbl = {1210.28025},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2010__46_6_a13/}
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Mesiar, Radko; Li, Jun; Pap, Endre. The Choquet integral as Lebesgue integral and related inequalities. Kybernetika, Tome 46 (2010) no. 6, pp. 1098-1107. http://geodesic.mathdoc.fr/item/KYB_2010__46_6_a13/