We repeat 2. 26 as

1. Theorem [P. 2].Let (X, T) be a- topological Markov chain. There is a unique T-invariant probability m such that hm (T) ≥ hμ (T) for all T-invariant Borel probabilities μ. m is Markov and is supported by X.

This result was proved in LP. 2J without the knowledge that Shannon had included a similar theorem in his 1948 paper [s. W.]. In this Appendix we interpret the relevant parts of Shannon's paper to compare his theorem with 1. We will show that Shannon proved:

2. Theorem. Let (X, T) be an aperiodic topological Markov chain. There is a Markov probability m on X such that hm (T) ≥ hμ (T) for all (compatible)Markov probabilities μon; X

Comparing 2 with 1, we notice that in 2 μ is allowed to run through only Markov probabilities. This (insignificant) restriction is natural since at the time Shannon wrote, entropy had not been defined in general - in Is. W. J he defined it, for the first time, in some special cases. More significantly, there is no explicit uniqueness statement in 2.

All of Shannon's work is in the setting of (a model of) a communication system and in fact he proves 2 for systems (superficially) more general than aperiodic topological Markov chains. We consider Shannon's communication system and state his theorem for such a system. Then we show that this theorem is equivalent to 2, and prove it by Shannon's method.