This entry is not precisely about some thing I stumbled upon but about some thing a that I wanted to learn for some time now, namely Young measures. Lately I had a several hour train ride and I had the book Kinetic Formulation of Conservation Laws with me.
While the book is about hyperbolic PDEs and their formulation as kinetic equation, it also has some pointers to Young measures. Roughly, Young measures are a way to describe weak limits of functions and especially to describe how these weak limits behave under non-linear functions, and hence, we start with this notation.
1. Weak convergence of functions
We are going to deal with sequences of function in spaces for some open bounded domain and some .
For the dual space of is with and the dual pairing is
Hence, a sequence converges weakly in to , if for all it holds that
We denote weak convergence (if the space is clear) with .
For the case one usually uses the so-called weak-* convergence: A sequence in converges weakly-* to , if for all it holds that
The reason for this is, that the dual space of is not easily accessible as it can not be described as a function space. (If I recall correctly, this is described in “Linear Operators”, by Dunford and Schwarz.) Weak-* convergence will be denoted by .
In some sense, it is enough to consider weak-* convergence in to understand what’s that about with Young measures and I will only stick to this kind of convergence here.
Example 1 We consider and two values . We define a sequence of functions which jumps between these two values with an increasing frequency:
The functions look like this:
To determine the weak limit, we test with very simple functions, lets say with . Then we get
Hence, we see that the weak-* limit of the (which is, by the way, always unique) has no other chance than being
In words: the weak-* limit converges to the arithmetic mean of the two values between which the functions oscillate.
2. Non-linear distortions
Now, the norm-limit behaves well under non-linear distortions of the functions. Let’s consider a sequence which converges in norm to some . That is, . Since this means that we see that for any boundedcontinuous function we also have and hence .
The same is totally untrue for weak-* (and also weak) limits:
Example 2 Consider the same sequence as in example~1which has the weak-* limit . As a nonlinear distortion we take which gives
Now we see
The example can be made a little bit more drastically by assuming which gives . Then, for every with we have . However, with such a we may construct any constant value for the weak-* limit of (take, e.g. , ).
In fact, the relation is only true for affine linear distortions (unfortunately I forgot a reference for this fact\dots).
It arises the question, if it is possible to describe the weak-* limits of distortions of functions and if fact, this will be possible with the notions of Young measure.
3. Young measures
In my understanding, Young measures are a method to view a function somehow a little bit more geometrically in giving more emphasis on the graph of the function rather than is mapping property.
We start with defining Young measures and illustrate how they can be used to describe weak(*) limits. In what follows we use for the Lebesgue measure on the (open and bounded) set . A more through description in the spirit of this section is Variational analysis in Sobolev and BV spaces by Attouch, Buttazzo and Michaille.
Definition 1 (Young measure) A positive measure on is called a Young measureif for every Borel subset of it holds that
Hence, a Young measure is a measure such that the measure of every box is determined by the projection of the box onto the set , i.e. the intersection on with which is, of course, :
There are special Young measures, namely these, who are associated to functions. Roughly spoken, a Young measure associated to a function is a measure which is equidistributed on the graph of .
Definition 2 (Young measure associated to ) For a Borel measurable function we define the associated Young measure by defining for every continuous and bounded function
It is clear that is a Young measure: Take and approximate the characteristic function by smooth functions . Then
The left hand side converges to while the right hand side converges to as claimed.
The intuition that a Young measure associated to a function is an equidistributed measure on the graph can be made more precise by “slicing” it:
Definition 3 (Slicing a measure) Let be a positive measure on and let be its projection onto (i.e. ). Then is sliced into measures , i.e. it holds:
- Each is a probability measure.
- The mapping is measurable for every continuous and it holds that
The existence of the slices is, e.g. proven in Variational analysis in Sobolev and BV spaces, Theorem 4.2.4.
For the Young measure associated to , the measure in Definition~3is and hence:
On the other hand:
and we see that slices into
and this can be vaguely sketched:
4. Narrow convergence of Young measures and weak* convergence in
Now we ask ourself: If a sequence converges weakly* in , what does the sequence of associated Young measures do? Obviously, we need a notion for the convergence of Young measures. The usual notion here, is that of narrow convergence:
Definition 4 (Narrow convergence of Young measures) A sequence of Young measures on converges narrowly to , if for all bounded and continuous functions it holds that
Narrow convergence will also be denoted by .
One may also use the non-continuous test functions of the form with a Borel set and a continuous and bounded , leading to the same notion.
The set of Young measures is closed under narrow convergence, since we may test with the function to obtain:
The next observation is the following:
Proposition 5 Let be a bounded sequence in . Then the sequence of associated Young measures has a subsequence which converges narrowly to a Young measure .
The proof uses the notion of tightness of sets of measures and the Prokhorov compactness theorem for Young measures (Theorem 4.3.2 in Variational analysis in Sobolev and BV spaces).
Example 3 (Convergence of the Young measures associated to Example 1) Consider the functions from Example~1and the associated Young measures . To figure out the narrow limit of these Young measures we test with a function with a Borel set and a bounded and continuous function . We calculate
i.e. the narrow limit of the Young measures is notthe constant function but the measure . This expression may be easier to digest in sliced form:
i.e. the narrow limit is something like the “probability distribution” of the values of the functions . This can be roughly put in a picture:
Obviously, this notion of convergence goes well with nonlinear distortions:
Recall from Example~1: The weak-* limit of was the constant function , i.e.
The observation from the previous example is in a similar way true for general weakly-* converging sequences :
Theorem 6 Let in with . Then it holds for almost all that
In other words: is the expectation of the probability measure .