Integral representation of coherent upper conditional prevision with respect to its associated Hausdorff outer measure: a comparison among the Choquet integral, the pan-integral and the concave integral

摘要

In this paper, it is proven that the natural extensions of a submodular coherent upper conditional probability, defined a class S properly contained in the power set of Omega, coincide on the class of all bounded and upper S-measurable random variables. Moreover, it is proven that a coherent upper conditional prevision can be represented as the Choquet integral, the pan-integral and the concave integral with respect to its associated Hausdorff outer measure hs and, denoted by S the s-field of all hs-measurable sets, all these integral representations agree with the Lebesgue integral on the class of all S-measurable random variables.