A Fixed-Point Farrago pp 131-144 | Cite as

# 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.

## References

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