Banach-Tarski Paradox

Motivation

Recall that by Vitali's Theorem, there is no function $\mu :2^{\mathbb {R} }\to [0,+\infty ]$ satisfying all three of the following properties:

• $\mu$ is countably additive,
• $\mu$ is translation invariant, and
• for each interval $(a,b)$, we have $\mu ((a,b))=b-a$.

This result can be easily be generalized to higher-dimensional Euclidean spaces.

To obtain a function that can reasonably measure Euclidean space, one might try to weaken some of the above properties. However, if we weaken only the requirement that $\mu$ be countably additive, namely by requiring that $\mu$ be merely finitely additive, we still run into problems in higher dimensions, as the Banach-Tarski Paradox illustrates.

The Banach-Tarski Paradox

In 1924, Stefan Banach and Alfred Tarski proved the following result:

Let $U$ and $V$ be arbitrary bounded open sets in $\mathbb {R} ^{n}$ for $n\geq 3$. There exist $k\in \mathbb {N}$