Jordan Decomposition of Signed Belief Functions

In this chapter we shall try to arrive at some decompositions into generalized or even into classical probabilistic belief functions inspired by, and similar to, the Jordan decomposition of signed measure. Here generalized belief function is a particular case of signed belief function supposing that the signed measure µ used when defining the belief function in question takes only non-negative values (possibly including +∞), so that µ is just what is called simply (σ-additive) measure in Halmos (1950). We shall also prove that generalizations of basic probability assignments, generated on P(S) with a finite S by signed belief functions, are also signed measures on the measurable space 〈 P (S), P (P (S))〉. The following well-known theorem will play the key role of an inspiration, but also as a technical tool for all our reasonings and constructions throughout this chapter.