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

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 Ω, 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 (Formula presented.) and, denoted by S the σ-field of all (Formula presented.) -measurable sets, all these integral representations agree with the Lebesgue integral on the class of all S-measurable random variables. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.