Upper and lower probabilities

Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the event's upper probability and the event's lower probability.
Because frequentist statistics disallow metaprobabilities, frequentists have had to propose new solutions. Cedric Smith and Arthur Dempster each developed a theory of upper and lower probabilities. Glenn Shafer developed Dempster's theory further, now known as Dempster–Shafer theory or Choquet.
More precisely, in the work of these authors, one considers in a power set,, a mass function 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 is infinite there can be such that there is no associated mass function. See p. 36 of Halpern. Probability measures are a special case of belief functions in which the mass function only assigns positive mass to the event space's singletons.
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 also relate to probabilistic logic: see Gerla.
Observe also that a necessity measure can be seen as a lower probability, and a possibility measure as an upper probability.