Young measure


In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. Young measures have applications in the calculus of variations and the study of nonlinear partial differential equations, as well as in various optimization. They are named after Laurence Chisholm Young who invented them, however, in terms of linear functionals already in 1937 still before the measure theory has been developed.

Definition

We let be a bounded sequence in, where denotes an open bounded subset of. Then there exists a subsequence and for almost every a Borel probability measure on such that for each we have in. The measures are called the Young measures generated by the sequence.

Example

For every minimizing sequence of subject to , the sequence of derivatives generates the Young measures. This captures the essential features of all minimizing sequences to this problem, namely developing finer and finer slopes of .