7
votes
1answer
152 views

Proof that there is no Banach-Tarski paradox in $\Bbb R^2$ using finitely additive invariant set functions?

I am wondering if anyone is familiar with the above topic? I have found a proof that it is possible to define a finitely additive invariant set function in $\mathbb{R}^2$ on the circle in Lax's book ...
0
votes
1answer
235 views

What (and how many) pieces does the Banach-Tarski Paradox break a sphere into?

Wikipedia says There exists a decomposition of the ball into a finite number of non-overlapping pieces which can then be put back together in a different way to yield two identical copies of the ...