Measure preserving maps

We covered over the previous four sections two specific examples of dynamical systems: the irrational rotation and the full shift on two symbols. In both examples were able to identify a measure that controlled the limiting behaviour of empirical averages. In this section we will begin to develop the abstract setting into which both examples fit: that of measure-preserving maps between finite measure spaces. This is the beginning of ergodic theory.

Invariant measures

A measure space (X,B,μ) is a probability space if μ(X)=1. We will only study measure-preserving maps of probability spaces in this course. The subject of infinite ergodic theory is concerned with the more general setting of measure-preserving transformations on infinite measure spaces.

Definition

Fix a probability space (X,B,μ). A measurable map T:XX is measure-preserving if μ(T1(B))=μ(B) for all BB.

Definition

Fix a measure space (X,B) and a measurable map T:XX. A measure μ on (X,B) is invariant for T if μ(T1(B))=μ(B) for all BB.

If one takes the view that:

then the assumption T is measure-preserving says roughly that events are no more or less likely from moment to moment. For example, if X=[0,1) and B=[0,log(2)) then μ(B) is the likelihood that a random point in X is less than log(2). The set T1(B)={xX:T(x)B} then consists of those points xX that will be in B one moment later, and if T is measure-preserving then it is as likely now as later that a random point will be smaller than log(2).

In drawing from the above analogy we must be careful: what is meant by "random" above is specified by μ and this puts the cart before the horse to some extent, because in applications one often has interesting dynamics in the form of the measurable map T:XX to begin with and seeks a measure that is invariant for T in order to apply the tools of ergodic theory, rather than begining with a probability space (X,B,μ) and attempt to understand its measure-preserving transformations.

We need, before going much further, to be able to deduce whether a given measurable map is measure-preserving. One way of achieving this is via the π-𝜆 theorem, which tells us we only need to check that two measures agree on special collections of sets in B known as π-systems.

Definition

Fix a set X. A collection DP(X) is a π-system if D is non-empty and whenever A,B belong to D one has AB in D as well.

Two particular π-systems will be important for us later on.

The π-system of intervals

Take X=[0,1). The collection {[a,b):0a<b1}{} is a π-system on X.

The π-system of cylinders

Take X={0,1}N. The collection {CX:C is a cylinder}{} is a π-system on X.

The following theorem - a special case of the π-𝜆 theorem suited to our needs - will simplify the task of verifying a measurable map is measure-preserving.

Theorem (The π-𝜆 theorem)

Fix a measurable space (X,B). Let μ and ν be measures on (X,B). If D is a π-system with σ(D)B and DDμ(D)=ν(D) then μ=ν.

Proof:

We will take this for granted.

Corollary

Fix a measure space (X,B,μ) and a π-system D with σ(D)B. A measurable map T:XX is measure-preserving if μ(D)=μ(T1(D)) for all DD.

Proof:

Since T is measurable the map ν:B[0,] defined by ν(B)=μ(T1(B)) is a measure on (X,B). Apply the π-𝜆 theorem.

Let us check that our two foundational examples are measure-preserving transformations.

Example (Irrational rotation)

Take X=[0,1) and fix αR. Define T:XX by T(x)=x+αmod1 for all xX. Let B be the Borel σ-algebra on X and let μ be the restriction of Lebesgue measure to B. Let us check that μ is an invariant measure for T. It suffices, by the π-𝜆 theorem, to check that μ([a,b))=μ(T1([a,b))) for all 0a<b1. But T1([a,b)) is either an interval of length ba or a union of two disjoint intervals whose lengths sum to ba. In either case, the total length is unchanged.

Example (Full shift)

Take X={0,1}N and define T:XX by (T(x))(n)=x(n+1) for all xX. Let B be the Borel σ-algebra on X and let μ be the (p,1p) coin measure on (X,B). Let us check that μ is an invariant measure for T. It suffices, by the π-𝜆 theorem, to check that μ([ϵ(1)ϵ(r)])=μ(T1([ϵ(1)ϵ(r)])) for all 0a<b1. But T1([ϵ(1)ϵ(r)])=[0ϵ(1)ϵ(r)][1ϵ(1)ϵ(r)] has the same measure as [ϵ(1)ϵ(r)] by direct calculation.

Poincaré recurrence

We finish with our first result about measure-preserving dynamical systems. It states that events BB with positive measure must happen infinitely often as one continues to iterate the dynamics.

Theorem

Let (X,B,μ,T) be a system. For every BB with μ(B)>0 there is nN with μ(B(Tn)1(B))>0.

Proof:

Suppose to the contrary that μ(B(Tn)1(B))=0 for all nN. Since μ is T invariant we deduce that μ((Tk)1(B)(Tk+n)1(B))=0 for all nN and therefore the sequence n(Tn)1(B) of sets in B is pairwise disjoint. But then we have for all NN that 1=μ(X)μ(BT1(B)(Tn)1(B))=Nμ(B) yielding a contradiction if one chooses N>μ(B)1.