ON MEASURABILITY AND REGULARITY W. JOHN WILBUR 1. Introduction. In its most abstract form the fundamental ques- tion concerning nonmeasurability is the question of the existence of measurable cardinals (see [7, p. 307]). If a cardinal fr$is not measur- able then a nontrivial totally finite measure cannot be defined on the power set of K. Thus for such a measure defined on subsets of {*$ we are assured of the existence of nonmeasurable sets. One reason why it might be desirable to take a less abstract approach to nonmeasur- ability is the following. While the existence of measurable cardinals has come to appear quite plausible (see [l, pp. 80, 81] and [7, p. 307]), for a more restricted class of measures it may yet always be possible to find nonmeasurable sets regardless of the cardinality of the underlying space. For example in [4, pp. 226, 227] it is shown that every nondiscrete locally compact group contains a subset that is nonmeasurable under every left translation invariant extension of left Haar measure. In this paper, assuming the axiom of choice, we shall demonstrate the existence of nonmeasurable sets for any nontrivial regular mea- sure defined on a locally compact Hausdorff space. In the case when the space is a metric space this problem has been solved (see [5, p. 146] or [4, pp. 218, 219]), but the methods do not seem amenable to direct generalization. We shall obtain our result by constructing a suitable continuous function from the space in question to the unit interval. 2. Preliminaries. Throughout, P will be a locally compact Hausdorff topological space. The families of compact and open sub- sets of P will be denoted by C and ft, respectively. The o--algebra generated by ft, i.e., the Borel sets, will be denoted by 03. If (P, 3TC, u) is a measure space, p will be called regular iff: (i) 03 C91Ï; (ii) m(C)<°°> for all CQQ; (Hi) u(0)=sup{p(C)\CQO, CQe} forallOGft; (iv) u(A)=inl{ß(0)\AQO, OQ(x\ for all ^G^IL This definition of regularity is taken from [5, p. 177]. It reduces to that given in [3, p. 224] when P is <7-compact and 03= 9TC. If /x is regu- lar, AQ^il, and p(AC\C)=0 for every CQQ we shall call A locally /¿-null (see [5, p. 122]). Received by the editors March 29, 1968. 741 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use