General form of Chebyshev type inequality for generalized Sugeno integral

Abstract

We prove a general form of Chebyshev type inequality for generalized upper Sugeno integral in the form of necessary and sufficient condition. A key role in our considerations is played by the class of m-positively dependent functions which includes comonotone functions as a proper subclass. As a consequence, we state an equivalent condition for Chebyshev type inequality to be true for all comonotone functions and any monotone measure. Our results generalize many others obtained in the framework of q-integral, seminormed fuzzy integral and Sugeno integral on the real half-line. Some further consequences of these results are obtained, among others Chebyshev type inequality for any functions. We also point out some flaws in existing results and provide their improvements. * Mathematics Subject Classification (2010): Primary 28A25, 28E10, Secondary 91B06, 60E05

On Null-Continuity of Monotone Measures

“…We point out that, in our discussions, we only consider the Sugeno integral, the Choquet integral, pan-integral, and concave integral. In fact, several results we obtained in Section 4 can be easily extended to the class of generalized Sugeno integrals (e.g., see [32,33]), such as, seminormed fuzzy integrals under the semicopula. For instance, Theorem 7 remains valid under the condition that semicopula has no zero divisors, and the convergence result of Proposition 12 is also true for generalized Sugeno integrals.…”

Generalized level measure based on a family of conditional aggregation operators

“…This paper deals with a study of the concept of the generalized level measure being a generalization of the level measure µ({x ∈ X : f (x) a}) using in many mathematical formulas such as the Choquet integral [8], the Sugeno integral [27], the seminormed fuzzy integral [5] and their generalizations [2,22,25] (see Sec. 2.3 for details).…”