Locally Coherent Rates of Exchange

A theory of coherence is formulated for rates of exchange between events. The theory can be viewed as a generalization of de Finetti's theory of coherence as well as the theory of conditional coherence. Coherent rates of exchange on a fixed Boolean algebra are in one-to-one correspondence with finitely additive conditional probability measures on the algebra. Results of Renyi and Krauss on conditional probability spaces are used to show that coherent rates of exchange are generated by ordered families of finitely additive measures, possibly infinite measures. This provides an interpretation of improper prior distributions in terms of coherence. An extension theorem is proved and gives a generalization of extension theorems for finitely additive probability measures.