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
![\displaystyle \begin{array}{rcl} \int_{[0,1]\times{\mathbb R}}\phi(x,y){\mathrm d}{\mu^{f_n}(x,y)} &= &\int_0^1\phi(x,f_n(x)){\mathrm d}{\mathfrak{L}(x)}\\ & = &\int_B\psi(f_n(x)){\mathrm d}{\mathfrak{L}(x)}\\ & \rightarrow &\mathfrak{L}(B)\frac{\psi(a)+\psi(b)}{2}\\ & = & \int_B\frac{\psi(a)+\psi(b)}{2}{\mathrm d}{\mathfrak{L}(x)}\\ & = & \int_{[0,1]}\int_{\mathbb R}\phi(x,y){\mathrm d}{\bigl(\tfrac{1}{2}(\delta_a+\delta_b)\bigr)(y)}{\mathrm d}{\mathfrak{L}(y)}. \end{array} \displaystyle \begin{array}{rcl} \int_{[0,1]\times{\mathbb R}}\phi(x,y){\mathrm d}{\mu^{f_n}(x,y)} &= &\int_0^1\phi(x,f_n(x)){\mathrm d}{\mathfrak{L}(x)}\\ & = &\int_B\psi(f_n(x)){\mathrm d}{\mathfrak{L}(x)}\\ & \rightarrow &\mathfrak{L}(B)\frac{\psi(a)+\psi(b)}{2}\\ & = & \int_B\frac{\psi(a)+\psi(b)}{2}{\mathrm d}{\mathfrak{L}(x)}\\ & = & \int_{[0,1]}\int_{\mathbb R}\phi(x,y){\mathrm d}{\bigl(\tfrac{1}{2}(\delta_a+\delta_b)\bigr)(y)}{\mathrm d}{\mathfrak{L}(y)}. \end{array}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cbegin%7Barray%7D%7Brcl%7D+%5Cint_%7B%5B0%2C1%5D%5Ctimes%7B%5Cmathbb+R%7D%7D%5Cphi%28x%2Cy%29%7B%5Cmathrm+d%7D%7B%5Cmu%5E%7Bf_n%7D%28x%2Cy%29%7D+%26%3D+%26%5Cint_0%5E1%5Cphi%28x%2Cf_n%28x%29%29%7B%5Cmathrm+d%7D%7B%5Cmathfrak%7BL%7D%28x%29%7D%5C%5C+%26+%3D+%26%5Cint_B%5Cpsi%28f_n%28x%29%29%7B%5Cmathrm+d%7D%7B%5Cmathfrak%7BL%7D%28x%29%7D%5C%5C+%26+%5Crightarrow+%26%5Cmathfrak%7BL%7D%28B%29%5Cfrac%7B%5Cpsi%28a%29%2B%5Cpsi%28b%29%7D%7B2%7D%5C%5C+%26+%3D+%26+%5Cint_B%5Cfrac%7B%5Cpsi%28a%29%2B%5Cpsi%28b%29%7D%7B2%7D%7B%5Cmathrm+d%7D%7B%5Cmathfrak%7BL%7D%28x%29%7D%5C%5C+%26+%3D+%26+%5Cint_%7B%5B0%2C1%5D%7D%5Cint_%7B%5Cmathbb+R%7D%5Cphi%28x%2Cy%29%7B%5Cmathrm+d%7D%7B%5Cbigl%28%5Ctfrac%7B1%7D%7B2%7D%28%5Cdelta_a%2B%5Cdelta_b%29%5Cbigr%29%28y%29%7D%7B%5Cmathrm+d%7D%7B%5Cmathfrak%7BL%7D%28y%29%7D.+%5Cend%7Barray%7D+&bg=ffffff&fg=000000&s=0&c=20201002)
We conclude:

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.
![\displaystyle \phi\circ f_n \rightharpoonup^* \tfrac{\phi(a)+\phi(b)}{2}\chi_{[0,1]}. \displaystyle \phi\circ f_n \rightharpoonup^* \tfrac{\phi(a)+\phi(b)}{2}\chi_{[0,1]}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cphi%5Ccirc+f_n+%5Crightharpoonup%5E%2A+%5Ctfrac%7B%5Cphi%28a%29%2B%5Cphi%28b%29%7D%7B2%7D%5Cchi_%7B%5B0%2C1%5D%7D.+&bg=ffffff&fg=000000&s=0&c=20201002)
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
.
Like this:
Like Loading...