Banach-tarski Decompositions Using Sets with the Property of Baire

Perhaps the most strikingly counterintuitive theorem in mathematics is the "Banach-Tarski paradox": A ball in R3 can be decomposed into finitely many pieces which can be rearranged by rigid motions and reassembled to form two balls of the same size as the original. The Axiom of Choice is used to construct the decomposition; the "paradox" is resolved by noting that the pieces cannot be Lebesgue measurable. In this paper, we solve a problem posed by Marczewski in 1930 by showing that there are paradoxical decompositions of the unit ball using pieces which have the property of Baire. We also prove related "paradoxes" which involve only open sets; these proofs are entirely constructive and make no use of the Axiom of Choice. One such result is the following: one can find a finite collection of disjoint open subsets of the unit ball which can be rearranged by suitable isometries to form a set whose closure is a solid ball of radius 1010. For both kinds of decomposition, the results are not limited to R3. In fact, we produce such decompositions for the sphere Sn (n ~ 2) and for the unit ball in Rn (n ~ 3), as well as related spaces. (See Theorem 2.6, among others.) It follows (Proposition 2.11) that, for n ~ 2, there is no isometry-invariant, finitely additive measure on the subsets of Sn (Rn+!) with the property of Baire which gives measure 1 to Sn (the unit ball in Rn+! ). For other equivalent forms of Marczewski's problem and for additional references, see Mycielski [6] or chapter 9 of Wagon [9]. From the above, standard arguments show that, if A and B are bounded subsets of Rn (n ~ 3) with the property of Baire and with non empty interior, then A can be decomposed into finitely many sets with the property of Baire which can be rearranged to form B (Corollary 2.7). All of these results are consequences of a theorem about decompositions of open sets which is quite general: it holds in any Polish space (complete separable