When do equidecomposable sets have equal measures?

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 countable equidecomposability. As an application, we prove that if $G$ is a discrete group of isometries of ${\mathbb {R}^n}$, then every two Lebesgue measurable, countably $G$-equidecomposable subsets of ${\mathbb {R}^n}$ have equal measures.