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Notational convention: In this post only, I will colour a statement red if it assumes the axiom of choice. (For the rest of the course, the axiom of choice will be implicitly assumed throughout.) $\diamond$

The famous Banach-Tarski paradox asserts that one can take the unit ball in three dimensions, divide it up into finitely many pieces, and then translate and rotate each piece so that their union is now two disjoint unit balls.  As a consequence of this paradox, it is not possible to create a finitely additive measure on ${\Bbb R}^3$ that is both translation and rotation invariant, which can measure every subset of ${\Bbb R}^3$, and which gives the unit ball a non-zero measure. This paradox helps explain why Lebesgue measure (which is countably additive and both translation and rotation invariant, and gives the unit ball a non-zero measure) cannot measure every set, instead being restricted to measuring sets that are Lebesgue measurable.

On the other hand, it is not possible to replicate the Banach-Tarski paradox in one or two dimensions; the unit interval in ${\Bbb R}$ or unit disk in ${\Bbb R}^2$ cannot be rearranged into two unit intervals or two unit disks using only finitely many pieces, translations, and rotations, and indeed there do exist non-trivial finitely additive measures on these spaces. However, it is possible to obtain a Banach-Tarski type paradox in one or two dimensions using countably many such pieces; this rules out the possibility of extending Lebesgue measure to a countably additive translation invariant measure on all subsets of ${\Bbb R}$ (or any higher-dimensional space).

In these notes I would like to establish all of the above results, and tie them in with some important concepts and tools in modern group theory, most notably amenability and the ping-pong lemma.  This material is not required for the rest of the course, but nevertheless has some independent interest.