TITLE: Strength of some claims about finitely additive measures on infinite sets?
QUESTION [6 upvotes]: 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).

REPLY [3 votes]: A quick note that (2) doesn't follow from ZF (assuming ZF is consistent) or even ZF+DC.  
(2)+Countable Choice for Finite Sets (CC(fin)) implies that every uncountable set has a non-principal (finitely additive, non-trivial) measure.  Let $\Omega$ be uncountable.  Let $\mu$ be as in (2).  Let $A = \{ x : \mu(\{ x \}) > 0 \}$.  The standard proof that a finite measure has only countable many atoms only uses CC(fin) (there are at most $n$ elements $x$ such that $\mu(\{ x \}) \in [1/n, 1/(n-1))$).  So $A$ is countable.  So $\mu(A)<1$.  Then $\nu(B) = \mu(B-A)$ defines a non-principal measure and is non-trivial since $\nu(\Omega)=1-\mu(A)>0$.
But Blass gave a model of ZF+DC with no non-principal measures ("A model without ultrafilters", Bull. Acad. Polon. Sci. 25 (1977), 329–331; I haven't read the paper, but Pincus and Solovay say that it proves this).  Since DC implies CC(fin), (2) is false in that model.
A different way to see that (2) doesn't follow from ZF is this.  Take a model where there is an infinite Dedekind-finite set but CC(fin) still holds (the Consequences of AC website lists a bunch of models where Form 10 is true and Form 9 is false).
Now if $\Omega$ is an infinite Dedekind-finite set and $\mu$ satisfies (2), then every point of $\Omega$ will have to be an atom of $\mu$ (or else $\mu(\Omega - \{ x \}) = \mu(\Omega)$).  But by CC(fin), the set of atoms is countable, which is impossible for a Dedekind-finite set.
I'm curious if (2) implies CC(fin).