A general type of belief structure and its inducing dual pair of belief and plausibility functions on infinite universes of discourse are first defined. Relationship between belief and plausibility functions in Dempser-Shafer theory of evidence and the lower and upper approximations in rough set theory is then established. It is shown that the probabilities of lower and upper approximations induced by an approximation space yield a dual pair of belief and plausibility functions. And for any belief structure there must exist a probability approximation space such that the belief and plausibility functions defined by the given belief structure are just respectively the lower and upper probabilities induced by the approximation space. Finally, essential properties of the belief and plausibility functions are examined. The belief and plausibility functions are respective a monotone Choquet capacity and an alternating Choquet capacity of infinite order.
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