Almost sure versions of the Riesz-Raikov strong law of large numbers

The classical theorem of Riesz and Raikov states that if a > 1 is an intege r and f is a function in L-1(R/Z), then the averages A(n)(a)f(x) = 1/n(f(x) + f(ax) + ... + f(a(n-1)x)) converge to the mean value of f over [0, 1] for almost every x in [0, 1]. I n this paper we prove that, for f in L-1 (R/Z), the averages A(n)(a)f(x) co nverge a.e. to the integral of f over [0, 1] for almost every a > 1. Furthe rmore we obtain convergence rates in this strong law of large numbers.