Ethan Bolker's theorem [1, 2] on the representation of functions resembling quotients of measures underlies the logic of decision [5] in the same way in which Hölder-like theorems on the representation of Archimedean ordered groups, semigroups, etc., are seen in [6] as underlying various foundational systems of measurement. In [3] and in chapter 9 of [1] Bolker applies his theorem to a system akin to that of [5], proving existence and uniqueness of probability and utility functions for preference rankings that satisfy certain axioms. Here I want to clarify the framework of [5] and bring Bolker's theorem to bear directly upon it. It should be noted that Bolker's theorem predated [5] and made it possible.

The logic of decision can be viewed as a theory with a binary relation term (“nonpreference”) as its only primitive (apart from the notation of set theory or, alternatively, of higher-order logic). A model of the theory is a pair (u, P) where P is a probability measure on a σ – field M of “events” (measurable sets) and u is an integrable function on the union, W, of M. A sentence of the theory is valid in a model iff true for all assignments to its free variables of sets in M, when for E ∈ M we define U(E) as the conditional expected utility.