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…)