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This paper introduces two types of preorders on the system of all non-empty sets of collections (i.e., the set of all decomposition systems) based on a fixed monotone measure μ. Each of them refines the previous two kinds of preorders of decomposition systems. By means of these two new preorders of decomposition systems we investigate the coincidences of decomposition integrals and that of superdecomposition integrals, respectively. The generalized integral equivalence theorem is shown in the general framework involving an ordered pair of decomposition systems. This generalized theorem includes as special cases all the previous results related to the coincidences among the Choquet integral, the concave (or convex) integral and the pan-integrals. Thus, a unified approach to the coincidences of several well-known decomposition and superdecomposition integrals is presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The convex integral introduced by Mesiar et al. (Superdecomposition integrals, Fuzzy Sets Syst. 259 (2015) 3–11) is a special superdecomposition integral related to convex functional. It can be regarded as the counterpart of concave integral, but it is not fully dual to the concave integral. This paper investigates some properties of the convex integral in detail. We show which of these results are valid both for the concave and convex integrals, which are dual to each other, and which are valid only for one of these two integrals. We also provide a necessary and sufficient condition for which two concave integrals (resp. convex integral) based on different monotone measures coincide with each other. [ABSTRACT FROM AUTHOR]

Decomposition integrals provide a framework for non-linear integrals that include Choquet, Shilkret, the PAN, and the concave integrals. All of these integrals found their applications in mathematics, notably in decision-making and economy. An important class of decomposition integrals is the class of integrals extending Lebesgue integral in the sense that the decomposition integral with respect to classical measures coincides with Lebesgue integral. In this paper, we consider finite spaces X only and discuss some necessary and sufficient conditions for this property. Also, some construction methods are given and exemplified. [ABSTRACT FROM AUTHOR]

Recently, in mathematical economics, Even and Lehrer introduced the decomposition integral (Even and Lehrer, 2014). In this paper, general versions of some well-known probabilistic inequalities for the decomposition integrals and the superdecomposition integrals are discussed that are still open for research. The main results of this paper generalize some previous results for particular integral inequalities obtained by several researchers in generalized probability theory. [ABSTRACT FROM AUTHOR]

Four different types of convolutions of aggregation functions (the upper, the lower, the super-, and the sub-convolution) are examined in the setting of both sub- and super-decomposition integrals defined on a finite space. Examples of the results of the paper are provided. As a by-product, the super-additive transformation of sub-decomposition integrals and the sub-additive transformation of super-decomposition integrals are fully characterized. Possible applications are indicated. [ABSTRACT FROM AUTHOR]

The decomposition integrals of set-valued functions with regards to fuzzy measures are introduced in a natural way. These integrals are an extension of the decomposition integral for real-valued functions and include several types of set-valued integrals, such as the Aumann integral based on the classical Lebesgue integral, the set-valued Choquet, pan-, concave and Shilkret integrals of set-valued functions with regard to capacity, etc. Some basic properties are presented and the monotonicity of the integrals in the sense of different types of the preorder relations are shown. By means of the monotonicity, the Chebyshev inequalities of decomposition integrals for set-valued functions are established. As a special case, we show the linearity of concave integrals of set-valued functions in terms of the equivalence relation based on a kind of preorder. The coincidences among the set-valued Choquet, the set-valued pan-integral and the set-valued concave integral are presented. [ABSTRACT FROM AUTHOR]

Four different types of convolutions of aggregation functions (the upper, the lower, the super-, and the sub-convolution) are examined in the setting of both sub- and super-decomposition integrals defined on a finite space. Examples of the results of the paper are provided. As a by-product, the super-additive transformation of sub-decomposition integrals and the sub-additive transformation of super-decomposition integrals are fully characterized. Possible applications are indicated.