This post develops necessary machinery to define relative notions of compact and weak mixing systems, seen in the previous posts. These notions are the basic objects used by Furstenberg in his structural theorem, which we shall investigate next month.
A basic concept in Mathematics, exploited in all areas, is Divide and Conquer: in order to analyse a mathematical object, one tries to investigate its smaller components which, in some specific sense, are simpler than the initial one and at the same time give sufficient information to the understanding of the larger structure.
An example of this philosophy occurs when one considers invariant subsets of continuous functions. For example, if such subset is minimal, then every point has dense orbit and, in particular, is recurrent. This simple idea gives a proof of Birkhoff theorem on the existence of recurrent points for continuous transformations defined in compact metric spaces.
Another example is Thurston Geometrization Theorem, in which a compact three-dimensional manifold is broken into the direct sum of prime three-manifolds. These prime three-manifolds are chosen among a set of 8 different geometric structures: spherical, euclidean, hyperbolic and so on.
In terms of measure theory, one has the notion of ergodicity, in which there are no non-trivial invariant sets. In this post, we will discuss another notion on measure theory, that of
1. Factors and extensions
Given two probability measure spaces and , we usually refer to a measure-preserving map as a spatial map from to such that
- it is measurable, that is, and
- it preserves measure, that is, for all .
In many situations, the underlying spaces and play a secondary role, merely providing the means for defining the -algebras and and the measures and . It will be advantageous to consider the boolean algebras (defined in ERT11) and and maps defined from to . With this, many problems regarding sets of zero measure and transformations not defined in every point are not anymore obstructions for the development of the theory and we only keep the essence of the object in study.
Definition 1 A homomorphism between two probability measure spaces is an injection such that
- , for ,
- , for , and
- , for .
In this situation, we say is a factor of or that is an extension of . If is a bijection, is called an isomorphism.
We remark that, provided properties (i) to (iii) hold, the map is necessarily an injection between the two boolean algebras. We leave this as an exercise to reader. Now, let’s see some
- Every measure space has a trivial factor: let with the trivial -algebra and probability . Then defined by is a homomorphism.
- Every -algebra defines the factor by the inclusion map .
Actually, the last example represents the general situation.
Proof: Define . The map is, by definition, an isomorphism.
In all cases of interest, the homomorphism is determined by a spatial map from to . In the spectral point of view, it induces an isometric inclusion from to that lifts maps from to by the composition map , for every . We denote this map by . By Lemma 2, is isometric to .
When , we have an inclusion between two closed Hilbert spaces and so it’s worth to infer who is the orthogonal projection from to .
2. Conditional expectation
Definition 3 If is a probability measure space and a -algebra, the condition expectation of a function with respect to is the unique -measurable map such that
If is a homomorphism, the conditional expectation of is the unique function for which . We denote by .
Remarks. For all properties to be proved, we sometimes assume, by means of Lemma 2, that the factor is defined by a sub--algebra.
Let us check that the above equality really defines a unique function. Consider the probability measure defined in by the identity
Clearly, is absolutely continuous with respect to the restriction of to . By Radon-Nikodym theorem, there is a unique -measurable density function such that
This proves (and so ) is well-defined and so, for every , there is a map
It is clearly a linear operator, which is actually bounded. For this last assertion, we make use of Jensen inequality.
For a proof, consult any basic book of probability. Applying Jensen inequality to the function , the boundedness of follows directly:
We thus have the
The good feature of Proposition 5, although its intrisic importance, is that many properties of conditional expectation can now be proved by means of approximation. We’ll usually do this when dealing with products of functions. The next proposition collects the main properties of the conditional expectation.
- whenever ,
- for every ,
- whenever , and
- If is another homomorphism, then .
Proof: (i). Clear.
(ii). Let . Given , consider the -measurable set . We have
implying that . As is arbitrary, .
(iii) is -measurable and so
(iv). First, assume for some . Then . For arbitrary ,
and so the equality is true in this case. For the general one, proceed by approximation.
(v). Take in relation (1).
(vi). Let . For every , we have
Until now, we didn’t say what the conditional expectation has to do with the orthogonal projection we wanted to investigate in the first time. Let denote the orthogonal projection of to . It is a simple fact that
Lemma 7 Restricted to , is equal to .
Proof: Assume and . For each , we have
by definition. As is dense in , we conclude the proof.
3. Measure-preserving systems
We now turn our attention to measure-preserving systems, which is our case of interest. In this section, we return to the usual notation of mathematical letter representing a mps. Consider two mps and .
Definition 8 A homomorphism from to is a homomorphism between the measure spaces such that . In this case, we say is a factor of or that is an extension of .
Roughly speaking, what happens is that, if one considers the fibers of elements of , the dynamics of T among the fibers is exactly the dynamics of . In other words, saying that is an extension of means that, after collapsing points into fibers, the remaining dynamics is that of . This is exactly we wanted: to decompose the dynamics into a simpler one. Again, let’s see some
- Every mps has a trivial factor: let be the trivial measure space defined in Example 1 and the spatial map in such that . The map given by , , is a homomorphism.
- Consider a mps and a -algebra such that . This defines another mps that is a factor of the initial one by the identity map .
- Given two mps and , the product mps (see ERT8) is an extension of both and via the projections.
- Given two measure spaces and , let be a mps and a measure-preserving transformations. They induce the skew-product system defined bythat is an extension of via the projection on the first coordinate.
As in Lemma 2, we have a classification of factors.
Lemma 9 For every homomorphism between mps, there exists a -algebra such that is isomorphic to .
Proof: Let . We have
and so is a mps. The same isomorphism between measure spaces defined in Lemma 2 represents an isomorphism between the mps and , due to the assumption that commutes and .
When one has a homomorphism between mps, conditional expectations have the additional property of being compatible with and , according to the
Proof: Fix . Applying a change of variable, we have
As , the uniqueness of guarantees the result.
4. Disintegration of measures
We now examine what takes place above the points , that is, on the fibers of a homomorphism between measure spaces: should contain as a sub-measure, in some sense. This is what happens when one considers the disintegration of with respect to .
For a proof of Rokhlin’s theorem, we refer the reader to Furstenberg’s book or to these lecture notes from Marcelo Viana.
We call each a conditional measure and write
for the disintegration of with respect to . By (i), these conditional measures capture the essence of the conditional expectation. It is a simple task to check that the disintegration is unique almost surely. This implies, among other things, the following
Proposition 12 Let be a homomorphism of mps and be the disintegration of with respect to . Then
which, by Proposition 10, is equal to
proving that .
5. Relative product of measure spaces
Suppose now that , , are measure spaces and , homomorphisms. One can form the fibre product space , where is defined via the disintegrations and by the integration formula
This measure space is, in a natural way, an extension of . Actually, if is the projection map on the first coordinate, then is a homomorphism.
Definition 13 The measure space is called the product of and relative to .
- is supported on the fibre-product set
- It is a simple task to check that if is defined in an analogous manner, then , which proves that is unambiguously an extension of . This also proves that, if is another extension, then the operation of relative product with respect to is associative:We denote this measure space simply as . See Furstenberg book for further detail.
From the probabilistic point of view, the relative product provides a probability space on which both -algebras of events and are represented and are independent conditioned upon . This is corroborated by the next
Proposition 14 If and , then
where the first expectation refers to the extension , the second to and the third to .
Proof: By Rokhlin theorem,