On bounded redundancy of universal codes

Consider stationary ergodic measures for which the difference between the expected length of a uniquely decodable code and the block entropy is asymptotically bounded by a constant. Using ergodic decomposition, it is shown that the number of such measures is less than the base of the logarithm raised to the power of that constant. In consequence, an analogous statement is derived for excess lengths of universal codes. The latter was previously communicated without proof.