
Joel H. Shapiro 

Abstract:
Notes for lectures I gave at the Portland State University Analysis
Seminar during the Fall term 2014. The notes paradoxical
decompositions, with special emphasis on the famous BanachTarski
Theorem, which asserts that the unit ball of Euclidean 3space can be
partitioned into finitely many pieces which can be reassembled, using
only rigid transformations (i.e. isometries of R^3) into two copies of the unit ball.

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