Assume ZF. Consider the claim:


(1) For any infinite set $\Omega$, there is a finitely additive probability measure $\mu:2^\Omega\to[0,1]$ with $\mu(A) = 0$ whenever $|A|<|\Omega|$.


Then (1) is implied by Hahn-Banach and the claim that the union of two sets of cardinality less than $|\Omega|$ has cardinality less than $|\Omega|$ when $\Omega$ is infinite. (Let $\scr B$ be the boolean algebra $2^\Omega$ and let $\scr I$ be the ideal of all subsets of $\Omega$ of smaller cardinality. HB is equivalent to the claim that every boolean algebra has a f.a. probability measure, so $\scr B/I$ has a f.a. probability measure, which lifts to a f.a. probability measure on $\scr B$ with the requisite property.) 


[Deleted Question 1 as it was predicated on something false.]


**Question 2:** Is anything known about the strength of (1)?


A related claim is:


(2) Every set has a f.a. probability measure on its powerset which is not supported on any set of smaller cardinality (i.e., if $|A|<|\Omega|$ then $P(A)<1$).


Clearly (1) implies (2) (at least in the infinite case, but the finite case of (2) is trivial). 


**Question 3:** Does (2) imply (1)?


**Updates:** The answers below so far show: (1) is equivalent to AC. ZF+DC is not strong enough to show (2). It's still open in this discussion whether (2) is equivalent to AC (I doubt it).



Strength of some claims about finitely additive measures on infinite sets?
Assertion (1) is equivalent to the Axiom of Choice. This is because it implies that for every infinite set $\Omega$, the set $\{A \subseteq \Omega : |A| \lt |\Omega|\}$ is an ideal. It follows from this that every infinite cardinal number $\mathfrak{m}$ is indecomposable — there are no cardinals $\mathfrak{p},\mathfrak{q} \lt \mathfrak{m}$ such that $\mathfrak{p} + \mathfrak{q} = \mathfrak{m}$ — which is known to be equivalent to the Axiom of Choice. (Note that this implies that any two cardinals are comparable for if $\mathfrak{p}$ and $\mathfrak{q}$ are incomparable then $\mathfrak{p} + \mathfrak{q}$ must be greater than both. The assertion that cardinals are linearly ordered is well-known to be equivalent to the Axiom of Choice.)


Assertion (2) is more promising and (2) does imply (1) when $|\Omega|$ happens to be indecomposable. To see this, suppose $\mu$ is a finitely additive measure such that $\mu(A) \lt 1$ whenever $|A| \lt |\Omega|$. Since $\mathcal{J} = \{A \subseteq \Omega : |A| \lt |\Omega|\}$ is an ideal, the directed limit $$\nu(X) = \lim\_{A \in \mathcal{J}} \frac{\mu(X-A)}{1-\mu(A)}$$ makes sense and the result is a finitely additive probability measure $\nu$ on $\Omega$ such that $\nu(A) = 0$ whenever $|A| \lt |\Omega|$.


