— 1. Introduction —
In the study of measurable dynamics, the basic object of study is a measure preserving system: a quadruple , where is a set, is a -algebra over , is a probability measure on and is a measurable map such that, for each , we have , where . If there exists a set such that and , then we can consider the measure preserving system , where , and . This system is a piece of the original system , and thus can be studied separately. If there is no such set then we say that the system is ergodic.
Analogous to the way primes are the building blocks of the integers, ergodic systems are the building blocks of measure preserving systems. When we want to prove certain statements about general measure preserving systems (such as Furstenberg’s multiple recurrence theorem, which is equivalent to the celebrated theorem of Szemeredi in arithmetic progressions) it might be useful to reduce them to the case when the system is ergodic. The tool that allows for this reduction is called the ergodic decomposition and can be compared to the fundamental theorem of arithmetic in our analogy between ergodic measure preserving systems and the prime numbers. I have used this method before on this blog, when presenting the ergodic theoretical proof of Roth’s theorem.
Before I state the theorem I need to establish some notation. Throughout this post, will usually denote a measurable space and will be a -measurable map. A probability is invariant under (or -invariant) if for all we have , equivalently if is a measure preserving system. The probability is ergodic if for every with we have or , equivalently if the system is ergodic.
The conclusion can be informally stated as , i.e., any -invariant probability is the convex combination of the ergodic measures .
In this post I will discuss and eventually give a full proof of the following weaker version of Theorem 2, which is in practice often strong enough.
— 2. Alternative approach —
Before giving a rigorous proof of Theorem 2 I will briefly describe an alternative way to think about this theorem. This can be formalized to give a full proof of the ergodic decomposition theorem. Let be a compact metric space and let be the Borel -algebra. Let be a map measurable with respect to . Let be the set of all -invariant probability measures over . To see that is non-empty, let be arbitrary and let be the probability measure over defined by . Since the space of probability measures over is weak compact, there exists some weak limit point for the sequence and it is not hard to see that .
Observe that is a convex set. In other words, if and then . Recall that an extreme point of a set in a linear space is a point such that whenever is written as a convex combination of points in and , then .
Proof: First let be an extreme point. Let be an invariant set such that (so we want to show that ). Let be the probability measure defined by for any . Since and is invariant under we have
Hence . If , then and is also -invariant. Thus we can create a -invariant measure defined by and we have . Since is an extreme point in this can’t happen, and hence as desired.
Now we prove the converse. Let be ergodic, and write with . For any -invariant set we have
Since and (strict inequalities!) we deduce that . This implies that both and are ergodic measures.
Now let be arbitrary. By the pointwise ergodic theorem there exists a set such that and for each we have
By the previous remark, also , and hence, again by the ergodic theorem, we have that for -almost every point in we have
Since the right hand side of the two previous displays is the same, we conclude that . Since was arbitrary, we conclude that , and then it follows that as well. Therefore is an extreme point in .
Denote by the subset of -ergodic measures. We now recall Choquet’s theorem, which, in this case, says that for any there exists some measure on (yes, this is a measure on a space whose points are measures!) such that . Note that this equality is between measures of , it can be made more precise by for every .
This conclusion follows the same spirit as Theorem 2 and is also called the Ergodic Decomposition. For most (if not all) applications, this is enough, although we get maybe a better understanding from the statement and proof of Theorem 2.
— 3. Examples —
I will try to give some intuition about Theorem 2 by exploring some examples first.
— 4. Proof of Theorem 2 —
Example 3 hints that in order to find all the ergodic measures of a given system, one should look at the invariant sets (observe, however, that not all -invariant sets give an ergodic measure: the set is invariant for the system of Example 3 and yet no ergodic measure has as its support).
Proof: Let and let . Then
and hence is closed under complements. Now let be a sequence of sets in and let . Then
and hence is closed under countable unions and therefore it is a -algebra.
Henceforth we will call the -algebra of invariant sets. It turns out that the ergodic measures that appear in Theorem 2 are the measures that arise from the disintegration of with respect to the -algebra of invariant sets.
Proof: Let . For and let . Then is -measurable and hence
for -a.e. .
Proof: We first prove that for almost every , is -invariant. More precisely, we will find a set with such that for every , the measure is -invariant. Let be a countable dense set. It suffices to show that for each there exists a set with and such that for every we have . Recall by the construction of conditional measure that , so we need to show that -a.e. But this follows from the following computation, which holds for each
We now show that almost every is ergodic. It suffices to show that for every , there exists a set with and such that for every ,
The pointwise ergodic theorem (see Theorem 2 in this post, or more precisely this stronger version) implies that the left hand side equals for every in a full -measure set. On the other hand, the left hand side is (where the conditional expectations are both taken with respect to the measure ). The desired conclusion now follows from Lemma 5.
Proof: } Let denote the invariant -algebra and let be the disintegration of with respect to , for some with . By Lemma 6 each of the measures is -invariant and -ergodic. By the properties of the disintegration of measures we have that for every , the map is -measurable, and hence it is -measurable and -invariant. Moreover, it follows from the properties of the disintegration of measures that
and this finishes the proof.