The greatest portion of papers dealing with the Dempster-Shafer theory consider the case when the basic universe is a finite set, so that all the numerical characteristics introduced and investigated in the D-S theory, including the believeability and plausibility functions as the most important ones, can be easily defined by well-known combinatoric formulas outgoing from a simple probability distribution (basic belief assignment, in the terms of D-S theory) on the power-set P(S) of all subsets of S. The obvious fact that these numerical characteristics can be equivalently defined also by appropriate set-valued random variables becomes to be of greater importance in the case when S is infinite. We investigate, in this paper, the case when the power-set P(S) over an infinite set S is equipped by a nonempty sigma-field L subset of P(P(S)) and when the belief and plausibility functions are defined by a set-valued random variable (i.e., L-measurable mapping) which takes a given probability space into the measurable space [P(S), L]. In general, the values of the two functions in question need not be defined for each subset T of S. Therefore, we define four extensions of these functions to whole the P(S), based on the well-known concepts of inner and outer measure, and investigate their properties; interesting enough, just one of them respect the philosophy of the D-S approach to uncertainty quantification and processing and keeps the properties possessed by believeability and plausibility functions defined over finite spaces.
