They are a quantification of the oscillation effect of the sequence in the limit.
Young measures have applications in the calculus of variations, especially models from material science, and the study of nonlinear partial differential equations, as well as in various optimization (or optimal control problems).
They are named after Laurence Chisholm Young who invented them, already in 1937 in one dimension (curves) and later in higher dimensions in 1942.
[1] Young measures provide a solution to Hilbert’s twentieth problem, as a broad class of problems in the calculus of variations have solutions in the form of Young measures.
[2] Young constructed the Young measure in order to complete sets of ordinary curves in the calculus of variations.
That is, Young measures are "generalized curves".
That is, the curve should be a tight jagged line hugging close to the x-axis.
No function can reach the minimum value of
is identically zero, but the pointwise limit
Instead, it is a fine mist that has half of its weight on
so in the weak sense, we can define
The definition of Young measures is motivated by the following theorem: Let m, n be arbitrary positive integers, let
if the limit exists (or weakly* in
are called the Young measures generated by the sequence
A partial converse is also true: If for each
, that has the same weak convergence property as above.
, the limit if it exists, will be given by[3] Young's original idea in the case
concentrated on graph of the function
) By taking the weak* limit of these measures as elements of
is the mentioned weak limit.
A Young measure (with finite p-moments) is a family of Borel probability measures
A trivial example of Young measure is when the sequence
and so the weak* limit is always a constant.
To see this intuitively, consider that at the limit of large
Take that captured part, and project it down to the x-axis.
should look like a fine mist that has probability density
), and perhaps after passing to a subsequence, the sequence of derivatives
generates Young measures of the form
This captures the essential features of all minimizing sequences to this problem, namely, their derivatives
will tend to concentrate along the minima