Approximations of upper and lower probabilities by measurable selections

A random set can be regarded as the result of the imprecise observation of a random variable. Following this interpretation, we study to which extent the upper and lower probabilities induced by the random set keep all the information about the values of the probability distribution of the random variable. We link this problem to the existence of selectors of a multi-valued mapping and with the inner approximations of the upper probability, and prove that under fairly general conditions (although not in all cases), the upper and lower probabilities are an adequate tool for modelling the available information. In doing this, we generalise a number of results from the literature. Finally, we study the particular case of consonant random sets and we also derive a relationship between Aumann and Choquet integrals.