Stipulate that the Axiom of Probabilistic Choice (APC) says that for every collection $\{ A_i : i \in I \}$ of non-empty sets, there is a function on $I$ that assigns to $i$ a finitely-additive probability measure $\mu_i$ on $A_i$.

Then Hahn-Banach (HB) implies APC, since HB is equivalent to the existence of a finitely-additive probability measure on every boolean algebra, and hence implies the existence of such a measure on the direct sum of the powerset algebras $P(A_i)$.

APC is non-trivial in that it implies the Banach-Tarski paradox and the existence of nonmeasurable sets (the proof Foreman and Wehrung use to show that HB implies Banach-Tarski works).

**Question:** Does APC imply HB?

(When I think about this, I find I keep on wanting to use Stone representation, but of course to do that would be to assume Boolean Prime Ideal, which is stronger than HB or APC.)

3more comments