Elementary Properties of Belief Functions

In this chapter we shall survey the most elementary properties of belief functions and some other characteristics derived from them (cf. Smets (1992) and the references mentioned in the end of the last chapter, e. g., for more detail). We shall suppose, throughout this chapter, that the probability space ‹Ω, A, P› and the measurable spaces ‹P (S), S› and ‹E, ε› are fixed, the dependence of belief functions on possible variations or modifications of these basic stones of our constructions will be investigated in some of the following chapters. We shall also suppose that if the state space S is finite, then the σ-field S is the maximal one, i. e., S = P(P(S)), so that the values m(A), bel*(A) (and bel(A), if m(φ) < 1) are defined for each A ⊂ S and obey the usual combinatoric definitions. The properties of belief functions concerning their possible combinations and actualizations will be investigated in the next chapter dealing with the Dempster combination rule.