Laurence Chisholm Young originally introduced the objects that are now called Young measures as “generalized curves/surfaces” in the late 1930s and early 1940s, see [280–282], to treat problems in the calculus of variations and optimal control theory that could not be solved using classical methods. His book [283] explains these objects and their applications in great detail (in particular, the “sailing against the wind” example from Section 1.6 is adapted from there). The theory of Young measures is now very mature and there are several monographs [57, 222, 235] that give overviews of the theory from different points of view.

In Chapter 7 we will consider relaxation problems formulated using Young measures. Further, as we will see in Chapters 8 and 9, Young measures provide a convenient framework to describe fine phase mixtures in the theory of microstructure. A second avenue of development—somewhat different from Young’s original intention—is to use Young measures as a *technical tool* only. This approach is in fact quite old and was probably first adopted in a series of articles by McShane from the 1940s [184–186]. There, the author first finds a Young measure solution to a variational problem, then proves additional properties of the obtained minimizing Young measure, and finally concludes that these properties entail that the generalized solution is in fact classical.

Several people contributed to Young measure theory from the 1970s onward, including Berliocchi & Lasry [39], Balder [24], Ball [27] and Kristensen [164], among many others. An important breakthrough in this respect was the characterization of the class of Young measures generated by sequences of gradients in the early 1990s by Kinderlehrer and Pedregal [157, 158], see Theorem 7.15. Their result places gradient Young measures in duality with quasiconvex functions (to be defined in the next chapter) via Jensen-type inequalities; another work in this direction is Sychev’s article [259]. Young measures can also be used to show regularity, see the recent work by Dolzmann & Kristensen [102]. Carstensen & Roubíček [56] considered numerical approximations.

Young measure theory was opened up to many new applications in the late 1970s and early 1980s, when Tartar [267, 268, 270] and Murat [209–211] developed the theory of compensated compactness and were able to settle many open problems in the theory of hyperbolic conservation laws; another important contributor here was DiPerna, see, for example, [98]. A key point of this strategy is to use the good compactness properties of Young measures to pass to limits in nonlinear quantities and then to deduce from pointwise and differential constraints on the generating sequences that the Young measure collapses to a point mass, corresponding to a classical function (so no oscillation phenomena occurred). Moreover, in this situation weak convergence improves to convergence in measure (or even in norm), hence the name compensated *compactness*. We discuss compensated compactness theory in Section 8.8.

The disintegration result from Theorem 4.4 is essentially contained in the result from probability theory that *regular conditional probabilities* exist, see, for instance, Theorem 89.1 in [234]. The result as stated also holds for vector-measures, see Theorem 2.28 of [15]. A stronger version of the Scorza Dragoni Theorem 4.5 can be found in Theorem 6.35 of [122]. Lemma 4.13 is a version of the well-known *decomposition lemma* from [125], another version is in [163].

In the case \(p=1\) the theory of (classical) Young measures is not very satisfactory and some important results such as Lemma 4.11 and Lemma 4.13 do not hold (see Problem 4.9 for a counterexample to Lemma 4.13 in the case \(p=1\)). The fundamental reason for this deficiency is that in \(\mathrm {L}^1\) norm-bounded sequences are not weakly precompact. A partial remedy can be found by weakening the notion of convergence to be employed in \(\mathrm {L}^1\). Then, one can use Chacon’s *biting lemma*:

More information on this topic can be found in [35] and Chapter 6 of [222]. We will return to the topic of Young measures generated by merely \(\mathrm {L}^1\)-bounded sequences and develop a much more satisfying theory in Chapter 12.