Conditional belief functions as lower envelopes of conditional probabilities in a finite setting

The aim is to provide a characterization of full conditional measures on a finite Boolean algebra, obtained as lower envelope of the extensions of a full conditional probability defined on another finite Boolean algebra. Such conditional measures are conditional belief functions defined by means of a generalized Bayesian conditioning rule relying on a linearly ordered class of belief functions. This notion of Bayesian conditioning for belief functions is compared with other well-known conditioning rules by looking for those conditional measures that can be seen as lower conditional probabilities.