Bollettino dell’Unione Matematica Italiana https://doi.org/10.1007/s40574-020-00220-1 Construction method of coherent lower and upper previsions based on collection integrals Serena Doria 1 · Radko Mesiar 2 · Adam Seliga 2 Received: 15 January 2020 / Accepted: 2 March 2020 © Unione Matematica Italiana 2020 Abstract Given a finite non-empty set Ω a new type of integral, named super-additive integral, is proposed to define coherent lower previsions on the class of all bounded functions. It is the extension to the class of all bounded random variables of a shift-invariant collection integral with respect to a collection D and a capacity μ. Related coherent upper previsions are also considered. Keywords Coherent lower prevision · Collection integral · Construction method 1 Introduction Coherent lower previsions are operators defined on the linear space of all bounded random variables satisfying the axioms of coherence [14]. Coherent lower probabilities are obtained as restrictions of coherent lower previsions to indicator functions but coherent lower previsions cannot be determined by coherent lower probabilities. In [14, sect. 2.7.3] an example is given to show that lower probabilities may not determine lower previsions, that is different lower previsions yield the same lower probability when they are restricted to events. Coherent upper previsions are obtained by coherent lower previsions by means the conjugacy property. A way to define coherent lower and upper previsions is to assess them by integrals. Different types of integrals are proposed in literature, based on different collections of subsets of Ω and with respect to different capacities. Their properties depend on the collection considered to define the particular integral and the capacity with respect to the integral is Dedicated to Professor Domenico Candeloro B Serena Doria serena.doria@unich.it Radko Mesiar radko.mesiar@stuba.sk Adam Seliga adam.seliga@stuba.sk 1 Department of Engineering and Geology, University G. d’Annunzio, Chieti-Pescara, Italy 2 Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05 Bratislava, Slovakia 123