PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 113, Number 3, November 1991 WHEN DO EQUIDECOMPOSABLE SETS HAVE EQUAL MEASURES? PIOTR ZAKRZEWSKI (Communicated by Andreas R. Blass) Abstract. Suppose that G is a group of bijections of a set X . Two subsets of X are called countably G-equidecomposable if they can be partitioned into countably many respectively G-congruent pieces. We present a simple combinatorial approach to problems concerning count- able equidecomposability. As an application, we prove that if G is a discrete group of isometries of R" , then every two Lebesgue measurable, countably G-equidecomposable subsets of R" have equal measures. 0. Introduction and preliminaries Suppose that we partition a Lebesgue measurable subset A of Rn into at most countably many pieces and rearrange them by rigid motions to form a partition of another measurable set B . The question is: Are the measures of A and B necessarily equal? It is perhaps one of the most surprising results of theoretical mathematics that the answer can be "no" even if the number of the pieces is finite, i.e. when the sets A and B are finitely equidecomposable. The famous theorem of Hausdorff, Banach, and Tarski states that in three-dimensional space R3, every two bounded sets with nonempty interior are finitely equidecomposable! (See [BT] or [W, p. 29].) On the other hand, Banach [B] proved that in the lower dimensions, i.e. in E and R , finitely equidecomposable measurable sets must have equal measure. It turns out that this is due to algebraic properties of the groups of isometries of the respective spaces. 0.1. Definition. Given a group G of isometries of Rn , let us say that sets A and B in K" are finitely (countably) G-equidecomposable if one can form a partition of B from a finite (at most countable) partition of A using motions restricted to G. It is a consequence of results of Tits, Wagon, and Mycielski that if G does not Received by the editors January 11, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 03E05; Secondary 28C10. Key words and phrases. Countably equidecomposable sets, selector of orbits, invariant extensions of Lebesgue measure. © 1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page 831 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use