Full conditional probabilities and the Axiom of Choice

Here’s a claim that turns out to be equivalent to the Axiom of Choice:

1. Given any non-empty set Ω and a collection M of [0,∞]-valued finitely additive measures on the powerset of Ω such that for any non-empty E ⊆ Ω there is a μ ∈ M with 0 < μ(E) < ∞, there is a full conditional probability P on the powerset of Ω definable in terms of the measures in M in the sense that for each non-empty E there is a μ ∈ M such that 0 < μ(E) < ∞ and for all A we have P(A|E) = μ(A∩E)/μ(E).

The easy direction of proof is from (1) to AC. Let M be the collection of all the finitely additive probability measures on Ω that assign probability one to some singleton. Clearly M has the desired properties. Now, for any non-empty E ⊆ Ω, there will be a μ ∈ M such that P(A|E) = μ(A∩E)/μ(E) and 0 < μ(E) < ∞. Thus, the point at which μ is concentrated must be in E. Moreover, it is clear that for each E, the measure μ must be unique. Let f(E) be the point at which μ is concentrated. This is a choice function for all subsets of Ω. Since Ω is an arbitrary non-empty set, we have AC.

The other direction follows by the method of proof of Lemma 3 here.