In Part I we saw that the main idea in the construction of a paradoxical decomposition of a set was to first get such a decomposition in a group acting on the set, and then transfer it to the set. An almost identical theme pervades the construction of invariant measures on a set X acted upon by a group G. If there is a finitely additive, left-invariant measure defined on all subsets of G, then it can be used to produce a finitely additive, G-invariant measure defined on all subsets of X. Such measures on X yield that X (and certain subsets of X) are not G-paradoxical.

It was von Neumann [246] who realized that such a transference of measures was possible, and he began the job of classifying the groups that bear measures of this sort. In this chapter we first study some properties of the class of groups having measures and show that it is fairly extensive, containing all solvable groups. We then give the important application to the case of isometries acting on the line or plane, obtaining the nonexistence of Banach-Tarski-type paradoxes in these two dimensions.

Definition 10.1. If, for a group G, µ is a finitely additive measure on P(G) such that µ(G) = 1 and µ is left-invariant (µ(gA) = µ(A) for g ∈ G, A ⊆ G), then µ will be called simply a measure on G.