On the Measure-theoretic Character of an Invariant Mean

0. Introduction. M. M. Day has shown that an abelian group has only one invariant mean if, and only if, the group is finite [3, Theorem 4, p. 534]. Knowing this we now ask to what extent the invariant means on a given abelian group are alike. In this paper we shall show that they all have the same "measure-theoretic character." An invariant mean on an abelian group G is a normalized, positive, translation-invariant linear functional defined on the space of all bounded complex-valued functions on G. There is a canonical way of associating with the invariant mean L a (unique) positive measure ß (which is invariant in a sense that will be made clear) defined on the Stone-Cech compactification of G (discrete topology). Using this one-to-one correspondence, we shall ascribe to the invariant mean L the measure-theoretic properties of ß. Thus the measure algebra of the invariant mean L is simply the measure algebra of the associated measure ß. Our main theorem (Theorem 2) is that the measure algebras of any two invariant means on G are isomorphic. A corollary extends this result to the measure algebras of invariant means on a subsemigroup of an abelian group. An essential lemma in the proof of the main theorem is derived in §2. We show that if m is the cardinality of G and is assumed to be infinite, then 2m is the dimension of the S2 space associated with an invariant mean on G. Because we believe this result to be of independent interest, we state it as Theorem 1. A corollary to Theorem 1 gives a similar result for invariant means on certain abelian semigroups. In particular, this corollary is applicable to the semigroup of positive integers. An invariant mean on this semigroup is called a Banach mean. The corollary shows that the S2 space associated with a Banach mean is not separable. The question of the separability of this S2 space was raised by P. Porcelli. As a final topic we shall discuss (in §4) some related problems. This discussion amounts to an exploration of further possibilities in the directions of Theorems 1 and 2, and we shall raise several questions in this connection.