Belief functions generated by signed measures

It is a well-known fact that the usual and already classical combinatorial definition of belief function over (the power-set of) a finite set can be generalized in such a way that belief function is defined by the quantile function of a set-valued (generalized) random variable defined over an abstract probability space. In this contribution we shall investigate a further stage of generalization resulting when the probability space in question is replaced by a measurable space equipped by a signed measure; signed measure is a sigma-additive set function which can take values also outside the unit interval, including the negative and infinite ones. An assertion analogous to the Jordan decomposition theorem for signed measures is stared and proved, according to which each signed belief function restricted to its finite values can be defined by a linear combination of two classical probabilistic belief functions, supposing that the basic set is finite. (C) 1997 Elsevier Science B.V.