Sufficient Conditions for the Existence of a Finitely Additive Probability Measure
1. Introduction. Suppose that we have given a qualitative relation, which is to be interpreted as "at least as probable as," over a family of events, then under what conditions can we construct an order preserving, additive probability meas-ure? A counterexample du'e to Kraft, Pratt, and Seidenberg [9] shows that certain obvious necessary conditions, which define what has been called a qualitative probability (see, e.g., p. 295 of [lo] or p. 32 of [Ill), arg insuficient to guarantee the existence of such a measure. Sufficient conditions for a finitely additive measure have been presented by de Finetti [4], Koopman [5], [B], [7], and Savage [ll]. They all involve, either explicitly or implicitly, the strong property that for each integer n the universal event can be partitioned into n events that are equally probable under the given ordering. Among other things, this forces the family of events to be infinite. The existence of countably additive measures has been studied by Villegas [14]. This paper provides still another axiomatization for the finitely additive representation ; it is of interest because it does not demand arbitrarily fine partitions into equally, likely events-in fact, finite models of the axioms exist. The proof is also of some inherent interest because it depends upon a theorem from the theory of extensive measurement. The parallel between the additivity of probability and of extensive measures has always been apparent, but to my knowledge no intimate connection between them has been previously shown. The reason is that the additivity of probability only holds for certain pairs of events-dis-joint ones-whereas in the classical theory of extensive measurement (see [2], [3], [12], and [13]) additivity holds over all pairs of entities without any restriction. However, Behrend [I] and Luce and Marley [9], in attempts to make extensive measurement more realistic, have shown that an additive representation can still be constructed when only certain pairs of entities can be concatenated. Their result, which is stated fully in Section 3, is used to prove the present probability theorem. 2. The axioms and representation theorem. A preliminary definition is needed. It states, in essence, what we shall mean by an event having, qualitatively , a probability equal to that of n disjoint copies of a given event. DEFINITION 1. Let X be a non-empty set, A a family of subsets of X that includes @ and that is closed under complementation and union, and …