Symmetric coherent upper conditional prevision defined by the Choquet integral with respect to Hausdorff outer measure

In a metric space symmetric fuzzy measures defined on the class of all subsets are introduced. Coherent upper conditional probabilties defined by Hausdorff outer measures are symmetric and distorted coherent upper conditional probabilities defined by Hausdorff outer measures with concave distortion are proven to be symmetric. Null events and symmetric events with respect to coherent upper conditional probabilities defined by Hausdorff outer measures are characterized. Coherent upper conditional prevision defined as Choquet integral with respect to Hausdorff outer measure is symmetric because it is invariant with respect to equimeasurable random variables.