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 .

Pick and define

I claim that as , . Indeed, to say that is locally convex means that it is topologized by a family of seminorms. Pick any seminorm . Then

where . This tends to zero as , and since was arbitrary we get the claim. Now as a result, any limit point of (and at least one exists by compactness) will be a fixed point. This proves the lemma.

**Interpretation of ergodicity **

Given , let denote the compact convex set of probability Borel measures on with respect to which is measure-preserving. We have shown is nonempty. The next result gives an interpretation of ergodicity.

Proposition 3is ergodic (i.e. is ergodic w.r.t. ) if and only if is an extreme point of .

Recall that an **extreme** point of a convex set in some vector space is one such that if and for , then . I.e. does not lie on any proper line segments contained in . Extreme points are interesting because of the theorem of Krein-Milman, which states that a compact convex set is the closed convex hull of its extreme points.

To prove the proposition, suppose is not ergodic, and let be a proper -invariant set. Then so is , and we have

In this way, we have expressed as a convex combination of two probability measures on , supported respectively on , and each of which is -invariant. So is not an extreme point.

Now suppose is ergodic and we can write

Then and are absolutely continuous with respect to , so by the Radon-Nikodym theorem there are with

I will show that almost everywhere.

Consider a constant and . I claim that is -invariant. Indeed,

and similarly

which one can put together to get

But have the same -measure (indeed, when added to they give the sets , which have the same measure). Since on one set and on the other, we have . By ergodicity, . Since this is true for every , is constant a.e. and . Same for .

March 31, 2010 at 6:05 am

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