Motivation
The definition of Young measures is motivated by the following theorem: Let m, n be arbitrary positive integers, let be an open bounded subset of and be a bounded sequence in [clarification needed]. Then there exists a subsequence and for almost every a Borel probability measure on such that for each we have
weakly in if the limit exists (or weakly* in in case of ). The measures are called the Young measures generated by the sequence .
A partial converse is also true: If for each we have a Borel measure on such that , then there exists a sequence , bounded in , that has the same weak convergence property as above.
More generally, for any Carathéodory function , the limit
if it exists, will be given by[2]
- .
Young's original idea in the case was to consider for each integer the uniform measure, let's say concentrated on graph of the function (Here, is the restriction of the Lebesgue measure on ) By taking the weak* limit of these measures as elements of we have
where is the mentioned weak limit. After a disintegration of the measure on the product space we get the parameterized measure .