This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, and logic. It unifies the results of contemporary research on the paradox and presents several new results including some unusual paradoxes in hyperbolic space. It also provides up to date proofs and discusses many unsolved problems.
Contents
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.
Reviews
"...a readable and stimulating book." Ward Henson, American Scientist
"...packed with fascinating and beautiful results." R.J. Gardner, Bulletin of the London Mathematical Society
"...this beautiful book is written with care and is certainly worth reading." Wlodzimierz Bzyl, Mathematical Reviews
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