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�n this paper we present a new class of decomposi�on integrals called the collec�on integrals. �rom this class of integrals we take a closer look on two special types of col-lec�on integrals� namely the chain integral and the minmax integral. �uperdecomposi�on �ersion of collec�on integral is also de�ned and the superdecomposi�on duals for the chain and the min-max integrals are presented. �lso� the condi�on on the collec�on that ensures the coincidence of the collec�on integral with the �e�esgue integral is presented. �astly� some computa�onal algorithms are discussed. Keywords: decomposi�on integrals� nonlinear integrals� computa�onal algorithmsExample 2.2. If H 1 consists of all singleton collections, we speak about the Shilkret integral [8], i.e.,Note that we use the following abbreviate notation min f (A) = ∧ {f (x) : x ∈ A}. If H 2 consists only of partitions of X then we speak about the Pan integral [9], i.e.,where Prt(X) denotes the set of all partitions on X. In case that H 3 is the class of all chains on X then the corresponding integral is the Choquet integral [1], i.e.,

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Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do not cover this important class of functionals on real random variables. In this paper, a new approach to the construction of coherent lower previsions acting on a finite space is proposed, exemplified and studied. It is based on special decomposition integrals recently introduced by Even and Lehrer, in our case the considered decomposition systems being single collections and thus called collection integrals. In special case when these integrals, defined for non-negative random variables only, are shift-invariant, we extend them to the class of all real random variables, thus obtaining so called super-additive integrals. Our proposed construction can be seen then as a normalized super-additive integral. We discuss and exemplify several particular cases, for example, when collections determine a coherent lower prevision for any monotone set function. For some particular collections, only particular set functions can be considered for our construction. Conjugated coherent upper previsions are also considered.

In this paper, we explore the use of aggregation functions in the construction of coherent upper previsions. Sub-additivity is one of the defining properties of a coherent upper prevision defined on a linear space of random variables and thus we introduce a new sub-additive transformation of aggregation functions, called a revenue transformation, whose output is a sub-additive aggregation function bounded below by the transformed aggregation function, if it exists. Method of constructing coherent upper previsions by means of shift-invariant, positively homogeneous and sub-additive aggregation functions is given and a full characterization of shift-invariant, positively homogeneous and idempotent aggregation functions on [0,∞[n is presented. Lastly, some concluding remarks are added.