Paradoxical Decompositions

Overview

In Chap. 10 we used the fixed-point theorem of Markov and Kakutani to show that every abelian group G is “amenable” in the sense that there is a G-invariant mean on the vector space B(G) of bounded, real-valued functions on G. We observed that existence of such a “mean” is equivalent to existence of a finitely additive probability “measure” on $\mathcal{P}(G)$, the algebra of all subsets of G, and we asked if every group turns out to be amenable. We showed that the free group F ₂ on two generators is not amenable by finding within F ₂ four pairwise disjoint subsets that could be reassembled, using only group motions, into two copies of F ₂.

Now we’ll see how this “paradoxical” property of F ₂, along with the Axiom of Choice, leads to astonishing results in set theory, most notably the famous Banach–Tarski Paradox, often popularly phrased as: Each (three dimensional) ball can be partitioned into a finite collection of subsets which can then be reassembled, using only rigid motions, into two copies of itself. Even more striking: given two bounded subsets of $\mathbb{R}^{3}$ with nonvoid interior, each can be partitioned into a finite collection of subsets that can be rigidly reassembled into the other. For this result the fixed-point theorem of Knaster and Tarski (Theorem 1.2) makes another appearance, this time to prove a far-reaching generalization of the Schröder–Bernstein Theorem.