Betting on continuous independent events.

de Finetti's Dutch book theorem explains why probability has to be additive on disjunctions of incompatible yes-no (Boolean) events. The theorem holds verbatim also for continuous events, or random variables, as formalized in Łukasiewicz logic. Independence, a subtler notion than incompatibility, has a probability-free definition: two events are logically independent if there is no logical relation between them. Our main result in this paper is that product is the only coherence preserving operation on books on logically independent sets of continuous (as well as yes-no) events. Thus, also the product law for stochastically independent sets of events is implicit in de Finetti's deeper notion of a coherent betting system. [ABSTRACT FROM AUTHOR]