This paper generalizes an earlier result by Shereshevsky defining Lyapunov exponents for one-dimensional cellular automata. The existence of an almost everywhere constant value for each of these exponents required particular conditions for the measure. In this paper, the existence is proved for a more suitable class of measures. Further, new exponents called the average Lyapunov exponents are defined. These are smaller than or equal to Shereshevsky’s Lyapunov exponents. An inequality involving the average Lyapunov exponents is proved. Examples to illustrate the results are included.