Stochastic independence with respect to upper and lower conditional probabilities deined by Hausdorff outer and inner measures

A new model of coherent upper conditional prevision is proposed in a metric space. It is defined by the Choquet integral with respect to the s-dimensional Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its dimension s. Otherwise if the conditioning event has Hausdorff outer measure in its dimension equal to zero or infinity it is defined by a 0-1 valued finitely, but not countably, additive probability. If the conditioning event has positive and finite Hausdorff outer measure in its dimension the coherent upper conditional prevision is proven to be monotone, comonotonically additive, submodular and continuous from below. Given a coherent upper conditional prevision the coherent lower conditional prevision is defined as its conjugate. In Doria (2007) coherent upper and lower conditional probablities are obtained when only 0-1 valued random variables are considered. The aim of this chapter is to introduce a new definition of stochastic independence with respect to coherent upper and lower conditional probabilities defined by Hausdorff outer and inner measures. A concept related to the definition of conditional probability is stochastic independence. In a continuous probability space where probability is usually assumed equal to the Lebesgue measure, we have that finite, countable and fractal sets (i.e. the sets with non-integer Hausdorff dimension) have probability equal to zero. For these sets the standard definition of independence given by the factorization property is always satisfied since both members of the equality are zero. The notion of s-independence with respect to Hausdorff outer and inner measures is introduced to check probabilistic dependence for sets with probability equal to zero, which are always independent according to the standard definition given by the factorization property. Moreover s-independence is compared with the notion of epistemic independence with respect to upper and lower conditional probabilities (Walley, 1991). The outline of the chapter is the following. In Section 2 it is proven that a conditional prevision defined by the Radon-Nikodymderivative may be not coherent and examples are given. 6