Conditional imprecise measures, independence, and products of interval probability measures

Recently, we introduced a coherent framework for interval-valued probability measures, providing a rigorous foundation for modeling imprecise probabilities and uncertainty in a natural and internally consistent manner. Building upon this foundation, the present paper develops a detailed analysis of the underlying theoretical structure, highlighting in particular some distinctive features of the theory of Intuitionistic Fuzzy Sets and introducing a new concept of conditional imprecise probability. In particular, we propose an updating rule for events that incorporates fundamental notions of statistical independence together with suitable product constructions for imprecise probability measures. Finally, these concepts are employed to define the notions of independent and incompatible random variables within the proposed framework, and to provide a refined interpretation of the classical umbrella problem.