Metric representations of a preference ordering

We prove that when we decompose the expected utility function inside of an mdimensional metric space we refer to a preference ordering based on the notion of distance. We prove that when we deal with a scale of measurable utilities we refer to a preference ordering based on the notion of distance. A contingent consumption plan is studied inside of an m-dimensional metric space because utility and probability are both subjective. The right closed structure in order to deal with utility and probability is a metric space in which we study coherent decisions under uncertainty having as their goal the maximization of the prevision of the utility associated with a contingent consumption plan.