Machine derived contents note: Part I. Paradoxical Decompositions, or the Nonexistence of Finitely Additive Measures: 1. Introduction -- 2. The Hausdorff Paradox -- 3. The Banach-Tarski Paradox: duplication spheres and balls -- 4. Locally commutative actions: minimizing the number of pieces in a paradoxical decomposition -- 5. Higher dimensions and non-Euclidean spaces -- 6. Free groups of large rank: getting a continuum of spheres from one -- 7. Paradoxes in low dimensions -- 8. The semi-group of equideomposability types -- Part II. Finitely Additive Measures, or the Nonexistence of Paradoxical Decompositions: 9. Transition -- 10. Measures in groups -- 11. Applications of amenability: Marczewski measures and exotic measures -- 12. Growth conditions in groups and supramenability -- 13. The role of the axiom of choice
