Construction method of coherent lower and upper previsions based on collection integrals

Given a finite non-empty set $$\varOmega $$ 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 $${\mathcal {D}}$$ and a capacity $$\mu $$. Related coherent upper previsions are also considered.