TITLE: Is $SO_2$ an amenable group?
QUESTION [9 upvotes]: In S. Wagon's "The Banach-Tarski Paradox," amenable groups are defined on p. 12 as follows:

[amenable] groups bear a left-invariant, finitely additive measure of total measure one that is defined on all subsets.

He defines $SO_2$ to be the group of rotations of the unit circle, which he has used to show that $S_1$, the unit circle, is $SO_2$-paradoxical (as an analogue to the usual non-measurable set defined in the interval $[0,1)$ ).  I am taking measure theory this term, but am not sure how to assign a measure to subsets of $SO_2$.  Thus, I am not really sure where to start in showing whether or not $SO_2$ is an amenable group.
When I look at the Wikipedia entry about amenable groups, I'm unable to make much more sense of the definition in the context of the material.
Is $SO_2$, the group of rotations of the unit circle, an amenable group?  If not, why (so that I may build an intuition for these objects)?

REPLY [6 votes]: An amusing point is that the proof that there is a measure on the Abelian group of rotations is generally done using the Hahn-Banach Theorem, and that requires the Axiom of Choice. So this means that the proof that a Banach-Tarski Paradox does NOT exist in the plane requires the same Axiom of Choice that yields the paradox in 3-space. 
But in fact one can eliminate (almost all of) AC from the proof that there is no BTP in the Euclidean plane.
Stan Wagon