PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 1, May 1975 CONVERGENCE OF AVERAGES OF POINT TRANSFORMATIONS M. A. AKCOGLU 'aNDA. DEL JUNCO Let (X, J, p) be a finite measure space. An invertible transformation of X which is measure preserving in both directions is called an automor- phism of X. Birkhoff's Ergodic Theorem states that if T is an automorphism of X then the sequence -^-r Z fir'x) n + 1 ""^ ¿=0 converges a.e. for each f £ L. = L,(X, J, p). This raises the following question. What are the necessary conditions on the matrix (a .) so that the sequence / (x) =Lfl -f(r~lx) converges a.e. for each / £ L, and for each automorphism r? The answer is not known. Spectral considerations would suggest, however, the following conjecture. If (a .) is such that the sequence of functions p (z) = X a .z~* is uniformly bounded and pointwise convergent on the unit circle \z\ = 1, then / converges a.e. In fact, recently an at- tempt has been made to prove this as a theorem [l]. In this note we would like to observe the following simple fact which shows that this conjecture is far from being correct. If r is a real number, let [r] denote the greatest integer which is less than or equal to r. Define a matrix (a .), n = 1, 2, • • •, as 1 . r . , if ?2 < i < L\n] + n, a . ni tyn] + 1 0 otherwise Then the a . certainly satisfy the hypotheses of the conjecture. However, we have the following result. Proposition. // r is an ergodic automorphism of a probability space (X, 3", p), then there is a set E such that there is a set B of positive measure on which X .a .yF(r~'x) fails to converge. Received by the editors July 23, 1974. AMS (MOS) subject classifications (1970). Primary 28A65, 47A35. 1 Research supported by NRC Grant A3974. Copyright © 1975. American Mathematical Society 265 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use