PROCEEDINGS OF THE AMERICANMATHEMATICAL SOCIETY Volume 121, Number 2, June 1994 THE EXISTENCE OF NONMEASURABLE SETS FOR INVARIANTMEASURES MARCIN PENCONEK AND PIOTR ZAKRZEWSKI (Communicated by Andreas Blass) Abstract. We prove that if G is a locally compact Polish group acting in a reasonable way on a set X , then for every countably additive, <r-finite, G- invariant measure on X there exist nonmeasurable sets. In particular, the latter is true when A" is a compact, metric, metrically homogeneous space, and G is the group of its isometries. 1. Introduction and preliminaries By a measure on a set X we mean a countably additive, nonzero function defined on a o -algebra of subsets of X and assuming values in [0, +00]. A measure is nonatomic if it vanishes on all singletons; it is cr-finite if X is a countable union of sets of finite measure. A measure on X is universal if it is defined on the rj-algebra 7?(X) of all subsets of X ; i.e., nonmeasurable sets do not exist. Suppose that a group G acts on a set X, by which we mean that there is a function (g, x) -> gx of G x X into X such that (i) for each g e G, x -* gx is a permutation of X, (ii) for all x e X and gx, g2 e G, gx(g2x) = (gx • g2)x. For any set AC X and any g e G we write g A = {gx : x e A} . A measure m: sé —> [0, +00] on X is G-invariant if gAe si and m(A) = m(gA) whenever A esé and g e G. For any x e X, Gx = {gx: g e G} and Gx = {g e G: gx = x) are the (/-orbit and the stabilizer of x, respectively. Although the existence of a set carrying a cr-finite, nonatomic, universal mea- sure cannot be proved in ZFC (see [1, Chapter 6]), direct arguments employing invariance properties of measures to show the existence of nonmeasurable sets within ZFC have always been of considerable interest (see, e.g., [2-4]). The Vitali construction of a Lebesgue nonmeasurable set is the prototype of such results. Received by the editors July 29, 1991 and, in revised form, September 29, 1992. 1991Mathematics SubjectClassification. Primary 03E15, 28A05; Secondary 28C10,03E05. Key words and phrases. Action by a locally compact Polish group, invariant cr-finite measure, nonmeasurable set. The main result of this paper was obtained while the second author was visiting the Technische Universität Berlin as a research fellow of the Alexander von Humboldt Foundation. ©1994 American Mathematical Society 0002-9939/94 $1.00+ $.25 per page 579 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use