Defining Coherent Upper Conditional Previsions in a General Metric Space Using Distinct Dimensional Fractal Outer Measures

This article presents a novel approach to constructing models of CUCP for both bounded and unbounded random variables in a metric space. The models are built upon the concept of different dimensional outer measures, and the relationships between these measures are thoroughly examined. By utilizing a dimensional measure, it becomes possible to determine the dimensional outer measure of the conditioning event, even in cases where the metric space is not second countable. This metric serves as an indicator of whether the conditioning event possesses a measure of zero, positive, finite, or infinite dimension within its specific context. In cases where the conditioning event exhibits a positive and finite-dimensional measure in its dimension, the coherent upper conditional forecast is established through the utilization of the Choquet integral. Additionally, this definition aligns with an expanded version of the monotone convergence theorem for non-linear integrals, ensuring the maintenance of equivalence among unbounded random variables with identical distributions. On the contrary, if the conditioning event has an outer measure dimensionally equal to 0 or \(+\infty \) , the coherent upper conditional forecast is delineated about a probability that is finitely but not countably additive, taking on values of 0 or 1.