Cedric Smith and Arthur Dempster each developed a theory of upper and lower probabilities.
More precisely, in the work of these authors one considers in a power set,
satisfying the conditions In turn, a mass is associated with two non-additive continuous measures called belief and plausibility defined as follows: In the case where
Probability measures are a special case of belief functions in which the mass function assigns positive mass to singletons of the event space only.
A different notion of upper and lower probabilities is obtained by the lower and upper envelopes obtained from a class C of probability distributions by setting The upper and lower probabilities are also related with probabilistic logic: see Gerla (1994).