Subgroups of sequences and paths

0. It is well known and easy to prove that if a measurable subgroup G of n-dimensional Euclidean space Rn has positive n-dimensional Lebesgue measure, then G = Rn. In the first section we give an analogue of this result for subgroups of real-valued sequences, where the space RX of all such sequences is given product normalized Gaussian measure. In the second section we use this result to sharpen and extend a zero-one law of Cameron and Graves [1]. 1. In this section, R is the set of real numbers, Rn is n-dimensional Euclidean space, and RX is the set of all sequences (xi, x2, * * * ) of real numbers. 63 is the class of Borel subsets of R, and GBn (G?0) is the corresponding product o-field over Rn (R??). We will use ,u to denote the Gaussian measure on (G3 mentioned above. We say that a subset of RX is measurable if it is in the completion of (Bo with respect to I THEOREM. If G is a measurable subgroup of RX, then either ,u(G) = 0 or j(G) = 1. LEMMA. Suppose that f and g are bounded functions on Ro, each measurable with respect to the ,A-completion of (3X. For each xERn, let gz be the function on RX defined by