Imprecise distribution function associated to a random set

Some different extensions to random sets of the most common parameters of a random variable share a common rationale: a random set represents the imprecise observation of a random variable, hence the generalized parameter contains the available information about the respective parameter of the imprecisely observed variable. Following the same principles, in this paper it is proposed a new definition of the distribution function of a random set. This definition is simpler in its formulation and it can be used in more general cases than previous proposals. The properties of the distribution function defined here are discussed: some issues about continuity, convergence of the images of the distribution function, monotonocity and measurability are studied. It is also stated that not all the information conveyed by the random set about the original probability measure (the probability measure induced by the imprecisely observed random variable) is kept by its distribution function. 2003 Elsevier Inc. All rights reserved.