In Chapter 12, we shall examine results for a large class of processes with memory, known as ergodic processes. We start this chapter with a quick review of the main concepts of ergodic theory, then state our main results: Shannon–McMillan theorem, compression limit, and asymptotic equipartition property (AEP). Subsequent sections are dedicated to proofs of the Shannon–McMillan and ergodic theorems. Finally, in the last section we introduce Kolmogorov–Sinai entropy, which associates to a fully deterministic transformation the measure of how “chaotic” it is. This concept plays a very important role in formalizing an apparent paradox: large mechanical systems (such as collections of gas particles) are on the one hand fully deterministic (described by Newton’s laws of motion) and on the other hand have a lot of probabilistic properties (Maxwell distribution of velocities, fluctuations, etc.). Kolmogorov–Sinai entropy shows how these two notions can coexist. In addition it was used to resolve a long-standing open problem in dynamical systems regarding isomorphism of Bernoulli shifts.

In this chapter, we provide a classical account of Kolmogorov–Sinai metric entropy for measure-preserving dynamical systems. We prove the Shannon–McMillan–Breimann Theorem and, based on Abramov's Formula, define the concept of Krengel's Entropy of a conservative system preserving a (possibly infinite) invariant measure.