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In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. 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[disambiguation needed] 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".[2]

Consider the problem of , where is a function such that , and continuously differentiable. It is clear that we should pick to have value close to zero, and its slope close to . That is, the curve should be a tight jagged line hugging close to the x-axis. No function can reach the minimum value of , but we can construct a sequence of functions that are increasingly jagged, such that .

The pointwise limit is identically zero, but the pointwise limit does not exist. Instead, it is a fine mist that has half of its weight on , and the other half on .

Suppose that is a functional defined by , where is continuous, then

so in the weak sense, we can define to be a "function" whose value is zero and whose derivative is . In particular, it would mean that .


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[3]


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 .

General definition

Let be arbitrary positive integers, let be an open and bounded subset of , and let . A Young measure (with finite p-moments) is a family of Borel probability measures on such that .


Pointwise converging sequence

A trivial example of Young measure is when the sequence is bounded in and converges pointwise almost everywhere in to a function . The Young measure is then the Dirac measure

Indeed, by dominated convergence theorem, converges weakly* in to

for any .

Sequence of sines

A less trivial example is a sequence

The corresponding Young measure satisfies[4]

for any measurable set , independent of . In other words, for any :

in . Here, the Young measure does not depend on and so the weak* limit is always a constant.

To see this intuitively, consider that at the limit of large , a rectangle of would capture a part of the curve of . Take that captured part, and project it down to the x-axis. The length of that projection is , which means that should look like a fine mist that has probability density at all .

Minimizing sequence

For every asymptotically minimizing sequence of

subject to (that is, the sequence satisfies ), 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 of the integrand .

If we take , then its limit has value zero, and derivative , which means .


  1. ^ Young, L. C. (1942). "Generalized Surfaces in the Calculus of Variations". Annals of Mathematics. 43 (1): 84–103. doi:10.2307/1968882. ISSN 0003-486X. JSTOR 1968882.
  2. ^ a b Balder, Erik J. "Lectures on Young measures." Cahiers de Mathématiques de la Décision 9517 (1995).
  3. ^ Pedregal, Pablo (1997). Parametrized measures and variational principles. Basel: Birkhäuser Verlag. ISBN 978-3-0348-8886-8. OCLC 812613013.
  4. ^ Dacorogna, Bernard (2006). Weak continuity and weak lower semicontinuity of non-linear functionals. Springer.