On a multiplication and a theory of integration for belief and plausibility functions

Belief and plausibility functions have been introduced as generalizations of probability measures, which abandon the axiom of additivity. It turns out that elementwise multiplication is a binary operation on the set of belief functions. If the set functions of the type considered here are defined on a locally compact and separable space X, a theorem by Choquet ensures that they can be represented by a probability measure on the space containing the closed subsets of X, the so-called basic probability assignment. This is basic for defining two new types of integrals. One of them may be used to measure the degree of non-additivity of the belief or plausibility function. The other one is a generalization of the Lebesgue integral. The latter is compared with Choquet's and Sugeno's integrals for non-additive set functions.