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 1 Let
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 2 Let
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 3
is 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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