Strong previsions of random elements

Let C be a class of arbitrary real random elements and P an extended real valued function on C. Two definitions of coherence for P are compared. Both definitions reduce to the classical de Finetti's one when C includes bounded random elements only. One of the two definitions (called strong coherence) is investigated, and some criteria for checking it are provided. Moreover, conditions are given for the integral representation of a coherent P, possibly with respect to a sigma-additive probability. Finally, the two definitions and the integral representation theorems are extended to the case where C is a class of random elements taking values in a given Banach space.