For convex subsets of vector spaces with finite dimension, it is clear that every element of the convex set may be written as a convex combination of the extremal elements. For example, every point in a triangle may be written as a convex combination of the vertices of the triangle. In view of the results in Section 4.3, it is natural to ask whether a similar property holds in the space of invariant probability measures, that is, whether every invariant measure is a convex combination of ergodic measures.

The ergodic decomposition theorem, which we prove in this chapter (Theorem 5.1.3), asserts that the answer is positive, except that the number of “terms” in this combination is not necessarily finite, not even countable. This theorem has several important applications; in particular, it permits the reduction of the proof of many results to the case when the system is ergodic.

We are going to deduce the ergodic decomposition theorem from another important result from measure theory, the Rokhlin disintegration theorem. The simplest instance of this theorem holds when we have a partition of a probability space (M,μ) into finitely many measurable subsets P1,…,PN with positive measure. Then, obviously, we may write μ as a linear combination

μ = μ(P1)μ1+· · ·+μ(PN)μN

of its normalized restrictions μi(E) = μ(E ⋂Pi)/μ(Pi) to each of the partition elements. The Rokhlin disintegration theorem (Theorem 5.1.11) states that this type of disintegration of the probability measure is possible for any partition P (possibly uncountable!) that can be obtained as the limit of an increasing sequence of finite partitions.

Ergodic decomposition theorem

Before stating the ergodic decomposition theorem, let us analyze a couple of examples that help motivate and clarify its content:

Example 5.1.1. Let f : [0,1] → [0,1] be given by f(x) = x2. The Dirac measures δ0 and δ1 are invariant and ergodic for f. It is also clear that x = 0 and x = 1 are the unique recurrent points for f and so every invariant probability measure μ must satisfy μ({0,1}) = 1. Then, μ = μ({0})δ0 + μ({1})δ1 is a (finite) convex combination of the ergodic measures.