Random intervals as a model for imprecise information

Random intervals constitute one of the classes of random sets with a greater number of applications. In this paper, we regard them as the imprecise observation of a random variable, and study how to model the information about the probability distribution of this random variable. Two possible models are the probability distributions of the measurable selections and those bounded by the upper probability. We prove that, under some hypotheses, the closures of these two sets in the topology of the weak convergence coincide, improving results from the literature. Moreover, we provide examples showing that the two models are not equivalent in general, and give sufficient conditions for the equality between them. Finally, we comment on the relationship between random intervals and fuzzy numbers.