PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 45, Number 2, August, 1974 ZERO-ONE LAWS FOR STABLE MEASURES R. M. DUDLEY1 AND MAREK KANTER ABSTRACT. For any stable measure p on a vector space, every measurable linear subspace has measure 0 or 1. 1. Introduction. It is known that for any Gaussian probability measure, a linear subspace has measure 0 or 1. This result has been extended to additive subgroups by Kallianpur [2]. Here we extend the zero-one law in a different direction, replacing "Gaussian" by "stable". We begin with some definitions. Definition. Let S be a vector space over R and let S be a (7-algebra of subsets of S. We call (S, S) a measurable vector space iff both the follow- ing hold: (a) addition is jointly measurable from S x S into S, (b) scalar multiplication is jointly measurable from R x S into S, fot completed Lebesgue measure À on R. Let S be a topological vector space and let J be the ff-algebra of Borel sets (generated by the open sets). Then if S is metrizable and sep- arable, (S, J ) is a measurable vector space, but it need not be so in general. If (S, S) is a measurable vector space and p and v are finite, counta- bly additive measures on S, then we have the convolution p * v defined as usual by (u*v)(A)= (px v)\{x, y): x + y e A\. For any (5-valued) random-variable Z, let its probability distribution (law), defined on o, be denoted by 5l(Z). Given any vector space S and c £ R, let 772 (x) = ex fot all x £ S, and 0s(x) = x + s for any s £ S. Definition. Given a measurable vector space (5, o), a probability Received by the editors October 9, 1973. AMS (MOS) subject classifications (1970). Primary 60B15; Secondary 60E05, 60 F 20, 28A40. Key words and phrases. Stable measure, strictly stable, zero-one law, measur- able vector space. This research was partially supported by National Science Foundation Grant GP-29072. Copyright © 1974, American Mathematical Society 245 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use