Hausdorff Outer Measures and the Representation of Coherent Upper Conditional Previsions by the Countably Additive Möbius Transform

This paper explores coherent upper conditional previsions, a class of nonlinear functionals that generalize expectations while preserving consistency properties. The study focuses on their integral representation using the countably additive Möbius transform, which is possible if coherent upper previsions are defined with respect to a monotone set function of bounded variation. In this work, we prove that an integral representation with respect to a countably additive measure is also possible, on the Borel σ-algebra, even when the coherent upper prevision is defined by the Choquet integral with respect to a Hausdorff measure, which is not of bounded variation. It occurs since Hausdorff outer measures are metric measures, and therefore every Borel set is measurable with respect to them. Furthermore, when the conditioning event has a Hausdorff measure in its own Hausdorff dimension equal to zero or infinity, coherent conditional probability is defined via the countably additive Möbius transform of a monotone set function of bounded variation. The paper demonstrates the continuity of coherent conditional previsions induced by Hausdorff measures.