Certain proofs and theorems involving equidecomposability would be much simplified if we could add sets. For instance, if X could literally be added to X to form 2X, then the fact that X is paradoxical could be stated simply as X = 2X. In fact, this can be done if we expand the group action appropriately so that multiple copies of X can be formed. This new context for discussing equidecomposability will allow us to state and prove theorems that otherwise would be very cumbersome. One of these is a cancellation law for equidecomposability that has several uses, the most important of which is its use in Tarski's theorem (9.2) relating paradoxical decompositions and invariant measures. Another application will be a proof that any two subsets of S2 with nonempty interior are equidecomposable using rotations. This expanded context for equidecomposability will also yield a simpler proof of Theorem 4.5 that a locally commutative action of a free non-Abelian group is paradoxical.

Definition 8.1. Suppose the group G acts on X. Define an enlarged action as follows. Let X* = X × N and let G* = {(g, π): g ∈ G and π is a permutation of N}, and let the group G* act on X* by (g, π)(x, n) = (g(x), π(n)). If A ⊆ X*, then those n ∈ N such that A has at least one element with second coordinate n are called the levels of A.