Characterization of coherent conditional probabilities as a tool for their assessment and extension

A major purpose of this paper is to show the broad import and applicability of the theory of probability as proposed by de Finetti, which differs radically from the usual one (based on a measure-theoretic framework). In particular, with reference to a coherent conditional probability, we prove a characterization theorem, which provides also a useful algorithm for checking coherence of a given assessment. Moreover it allows to deepen and generalise in useful directions de Finetti's extension theorem (dubbed as ''the fundamental theorem of probability''), emphasising its operational aspects in many significant applications.