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 is a probability space if . 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 . A measurable map is measure-preserving if
for all .
Definition
Fix a measure space and a measurable map . A measure on is invariant for if
for all .
If one takes the view that:
- each iterate of advances time by one unit, during which each point is transported to the new location ;
- each set describes a property that points in may or may not possess;
- assigns to sets the likelihood that a random point in has the property described by ;
then the assumption is measure-preserving says roughly that events are no more or less likely from moment to moment. For example, if and then is the likelihood that a random point in is less than . The set
then consists of those points that will be in one moment later, and if is measure-preserving then it is as likely now as later that a random point will be smaller than .
In drawing from the above analogy we must be careful: what is meant by "random" above is specified by 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 to begin with and seeks a measure that is invariant for in order to apply the tools of ergodic theory, rather than begining with a probability space 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 known as π-systems.
Definition
Fix a set . A collection is a π-system if is non-empty and whenever belong to one has in as well.
Two particular π-systems will be important for us later on.
The π-system of intervals
Take . The collection
is a π-system on .
The π-system of cylinders
Take . The collection
is a π-system on .
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 . Let and be measures on . If is a π-system with and
then .
Proof:
We will take this for granted. ▮
Corollary
Fix a measure space and a π-system with . A measurable map is measure-preserving if
for all .
Proof:
Since is measurable the map defined by is a measure on . Apply the π-𝜆 theorem. ▮
Let us check that our two foundational examples are measure-preserving transformations.
Example (Irrational rotation)
Take and fix . Define by
for all . Let be the Borel σ-algebra on and let be the restriction of Lebesgue measure to . Let us check that is an invariant measure for . It suffices, by the π-𝜆 theorem, to check that
for all . But is either an interval of length or a union of two disjoint intervals whose lengths sum to . In either case, the total length is unchanged.
Example (Full shift)
Take and define by
for all . Let be the Borel σ-algebra on and let be the coin measure on . Let us check that is an invariant measure for . It suffices, by the π-𝜆 theorem, to check that
for all . But
has the same measure as by direct calculation.
Poincaré recurrence
We finish with our first result about measure-preserving dynamical systems. It states that events with positive measure must happen infinitely often as one continues to iterate the dynamics.
Theorem
Let be a system. For every with there is with .
Proof:
Suppose to the contrary that for all . Since is invariant we deduce that
for all and therefore the sequence of sets in is pairwise disjoint. But then we have for all that
yielding a contradiction if one chooses . ▮