Up until now, we have concentrated on a transformation of a *fixed* measure space. We now take a different approach: is fixed, and we look for appropriate measures (on a fixed -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 1Let be a continuous transformation of the compact metric space . Then there exists a probability Borel measure on with respect to which is measure-preserving.

Consider the Banach space of continuous and the dual , which, by the Riesz representation theorem, is identified with the space of (complex) Borel measures on . The positive measures of total mass one form a compact convex subset of in the weak* topology by Alaoglu’s theorem. Now, induces a transformation of : . The adjoint transformation of is given by , where for a measure , . We want to show that has a fixed point on ; 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 2Let be a compact convex subset of a locally convex space , and let be the restriction of a continuous linear map on . Then has a fixed point in . (more…)