The Anscombe and Aumann double relation approach for defining subjective probabilities and utilities in terms of a person's preferences is generalized for the case in which the set of states of the world is unrestricted (finite or not). A monotone continuity condition enables us to prove $\sigma$-additivity; the necessity of this condition is also proved if our other assumptions hold. Although the single relation approach used by Fishburn appears to be more elegant, the present approach has the advantage of showing how the subjective probabilities arise.
"On Subjective Probability and Expected Utilities." Ann. Math. Statist. 43 (3) 928 - 933, June, 1972. https://doi.org/10.1214/aoms/1177692556