Up until now, we have concentrated on a transformation ${T}$ of a fixed measure space. We now take a different approach: ${T}$ is fixed, and we look for appropriate measures (on a fixed ${\sigma}$-algebra).  First, we will show that this space is nonempty.  Then we will characterize ergodicity in terms of extreme points.

This is the first theorem we seek to prove:

Theorem 1 Let ${T: X \rightarrow X}$ be a continuous transformation of the compact metric space ${X}$. Then there exists a probability Borel measure ${\mu}$ on ${X}$ with respect to which ${T}$ is measure-preserving.

Consider the Banach space ${C(X)}$ of continuous ${f: X \rightarrow \mathbb{C}}$ and the dual ${C(X)^*}$, which, by the Riesz representation theorem, is identified with the space of (complex) Borel measures on ${X}$. The positive measures of total mass one form a compact convex subset ${P}$ of ${C(X)^*}$ in the weak* topology by Alaoglu’s theorem. Now, ${T}$ induces a transformation of ${C(X)}$: ${f \rightarrow f \circ T}$. The adjoint transformation of ${C(X)^*}$ is given by ${\mu \rightarrow T^{-1}(\mu}$, where for a measure ${\mu}$, ${T^{-1}(\mu)(E) := \mu(T^{-1}E)}$. We want to show that ${T^*}$ has a fixed point on ${P}$; then we will have proved the theorem.

There are fancier methods in functional analysis one could use, but to finish the proof we will appeal to the simple

Lemma 2 Let ${C}$ be a compact convex subset of a locally convex space ${X}$, and let ${T: C \rightarrow C}$ be the restriction of a continuous linear map on ${X}$. Then ${T}$ has a fixed point in ${C}$. (more…)