On equivalence of decomposition integrals based on different monotone measures.

This paper investigates the coincidences of decomposition integrals based on different monotone measures. We propose the concept of total balance of a monotone measure related to decomposition integral and by means of its characteristics show a version of equivalence theorem of decomposition integrals. A binary relation on the set of monotone measures is introduced, which is a preorder related to a fixed decomposition system. Another version of the equivalence theorem of decomposition integrals is shown in the framework concerning preorder of a pair of monotone measures. As special cases, we present a necessary and sufficient condition that two pan-integrals (resp. concave integrals) with respect to different monotone measures coincide. By using the characteristics of total balance of monotone measures, it is shown that Chebyshev's Inequality holds for a decomposition integral if and only if the considered decomposition system is complete.(c) 2022 Elsevier B.V. All rights reserved.