Integrated Likelihood in a Finitely Additive Setting

Without a clear, precise and rigorous mathematical frame, is the likelihood "per se" a proper tool to deal with statistical inference and to manage partial and vague information? Since (as Basu puts it) "the likelihood function is after all a bunch of conditional probabilities", a proper discussion of the various extensions of a likelihood from a point function to a set function is carried out by looking at a conditional probability as a general non-additive "uncertainty" measure P(E| • ) on the set of conditioning events.