A Note on Invariant Finitely Additive Measures

We show that under certain general conditions any finitely additive measure which is defined for all subsets of a set X and is invariant under the action of a group G acting on X is concentrated on a G-invariant subset Y on which the G-action factors to that of an amenable group. The result is then applied to prove a conjecture of S. Wagon about finitely additive measures on spheres. It is well known that if G is an amenable group acting on a set X then there exist plenty of G-invariant finitely additive probability measures on (X, ^ß(X)) where ÍP(X) is the class of all subsets of X (cf. [3] for details). However, such measures may fail to exist when G is nonamenable. In [6] S. Wagon conjectured that if G is a group of isometries of Sn, the n-dimensional sphere, such that for any Ginvariant subset Y, the group {g/Y | g G G} of restrictions of elements of g to Y is nonamenable, then there does not exist any G-invariant finitely additive probability measure on (Sn,<#(Sn)): In this note we establish the above-mentioned conjecture. Further, we formulate a condition on actions of (abstract) groups, involving the isotropy subgroups and fixed point sets, which implies similar assertions in a more general situation (cf. Theorem 1.1). The condition holds for actions of subgroups G of any compact Lie group 0 acting on homogeneous spaces of 0. It also holds for actions of subgroups G of algebraic R-groups <£> acting on homogeneous spaces of 0 by algebraic Rsubgroups. Thus, in all these cases we are able to conclude that G-invariant finitely additive probability measures (defined for all subsets) are concentrated on invariant sets on which the action factors to that of an amenable quotient of G (cf. §2). A particular consequence is that if G is a nonamenable subgroup of GL(n+ 1, R) acting irreducibly on Rn+1, then for the natural G-actions on R"+1 — (0), 5n or Pn, there exist no invariant finitely additive probability measures (cf. Corollaries 2.2 and 2.4). We recall that by a theorem of A. Tarski [5] this is equivalent to existence of paradoxical decompositions for the action (cf. [6 and 7] for motivation and some results in that direction). In some of the G-actions discussed above, e.g. G a group of isometries of 5TM or G a subgroup of a compact Lie group 0 acting on a homogeneous space of 0, there exists a natural countably additive G-invariant probability measure defined on the class of Borel subsets. We prove that the measure extends to a G-invariant finitely additive measure defined on all subsets if and only if G is amenable (cf. Theorem Received by the editors January 10, 1984. 1980 Mathematics Subject Classification. Primary 28A70.