Banach Tarski Paradox in the Plane
Not quite sure that this is the right place for this question, but I couldn't find a topology or functional analysis section anywhere so here goes..
Basically, I have been given an assignment to prove that there is no Banach-Tarski paradox in R^2. I have decided to take the approach of first showing that there exists a nonnegative, finitely additive set function for all subsets of the unit circle that is invariant under rotations, up to here everything seems clear and I have been able to follow the proofs given in Lax's book "Functional analysis" which using the Hahn-Banach Theorem. Apparently there is a way of showing that the existence of such a function immediately proves that the Banach-Tarksi doesn't exist in R^2 but I don't see it.. Is the proof more complicated than I am thinking? Could someone please recommend a place where I can find a proof of this or at least offer me an explanation of why it works? Cheers