| Description |
This paper explores the Banach-Tarski paradox as well as the related notions of amenability and Følner sequences. Informally, the paradox states that one ball may be broken into finitely many pieces such that these pieces reassemble into two balls identical to the first one. To prove this paradox, we introduce the notion of equidecomposability and paradoxical decompositions. We then define an amenable group as one having a full, left-invariant, finitely-additive probability measure, and relate this notion back to paradoxical decompositions. As amenable groups are sometimes defined as groups which contain Følner sequences, we also discuss this notion and prove its equivalence to our definition of amenability. Finally, we use these tools to provide a proof of three different theorems which make up the Banach-Tarski paradox. |
