In their paper Sur les decompositions denombrables Banach and Tarski obtained a result which can be restated as follows: A necessary and sufficient condition that two Lebesgue measurable, euclidean sets a and a', shall have equal measure is that there exist null sets n < a and n' < a',such that a−n and a'−n' are the respective unions of sequences of disjoint measurable sets, of which corresponding sets are congruent.
If we then identify sets differing on only a null set, there exists a class of transformations on the measurable sets such that two sets are of equal measure if and only if they correspond under some transformation of that class. Then just as the elementary notion of the volume of an n-dimensional interval is generalized to that of any measure function on a Bore1 field, so the equally elementary notion of equality of volume, defined for these figures by the relation congruence, can be generalized to the notion of equality of measure defined for some field of sets by a suitable class of transformations. This is done as follows:
In Part I, we consider a complemented, distributive σ-lattice M with a zero element, and a class of Φ-isomorphisms on the principal ideals of M. The lattice M is to be taken to correspond to a family of measurable sets modulo the null sets, and Φ as a semi-group of measure-preserving transformations. Then Φ generates an equivalence relation a~b between elements of M which is countably additive and hereditary in the sense that a~b, a' < a imply that there exists a b ' < b such that a'~b'. It is then shown for the bounded elements of M,that is, those which are not equivalent to any proper subelement, that the relation a~b is also preserved by subtraction and by taking limits of monotonic sequences. It may be remarked that these latter results yield an independent proof of the theorem of Banach and Tarski. A construction is also given leading to a definition of a complete measure. In Part II, the measure of an element of M is defined as the totality of its equivalent elements, that is, of images under Φ; and a partial ordering and an operation of addition are defined in the set of measures. A necessary and sufficient condition is then given that the set of measure values be order and addition isomorphic to a set of positive numbers. Such an isomorphism is necessarily unique, up to a multiplicative constant. These results will be analyzed in detail in another paper. In Part III, a sufficient condition is given that, for a family of sets upon which a numerical measure is defined, two sets of equal measure shall correspond under some measure-preserving transformation, and so that the given measure shall be derivable by the procedure of Part II. It turns out that this is the case for a measure on a separable Bore1 field, provided there are no minimum sets of positive measure. However, a trivial generalization of the theory of Part II can be seen to be applicable to any measure function.