The theory of generalized Young measures started with the article [99] by DiPerna and Majda, who were interested in turbulence concentrations in fluid dynamics (their measures, however, described \(\mathrm{L}^2\)-concentrations). The \(\mathrm{L}^1\)-framework was first introduced by Alibert & Bouchitté [6] and then developed into the form presented here in [168], partly inspired by the approaches to classical Young measures in the papers [39, 258]. Many authors have developed the theory further, a selection of some relevant papers is [121, 155, 171, 172, 258, 265].

In our setting we only work with the sphere compactification of \(\mathbb {R}^N\). This is somewhat implicit, but the basic idea is explained in the opening remarks of this chapter, also see Problem 12.2. The theory can be extended to much more general target spaces and compactifications. For instance, \(\mathbb {R}^N\) may be replaced by a Banach space *X* with the analytic Radon–Nikodým property, i.e., the validity of the Radon–Nikodým theorem for *X*-valued measures, for this see [24]. Further, we may employ another compactification of \(\mathbb {R}^N\), for instance, the compactification generated by a separable, complete ring of continuous bounded functions, see Section 4.8 in [119] and also [60], or even the Stone–Čech compactification \(\beta \mathbb {R}^N\). Finally, \(\varOmega \) may be replaced by a general finite measure space. Such generalizations are discussed in [6, 24, 57, 99, 172].

A generalized Young measure

\(\nu \in \mathbf {Y}^\mathscr {M}(\varOmega ;\mathbb {R}^N)\) may also be described in the spirit of Berliocchi–Lasry [39] as follows:

Then,

$$ \nu (D \times \mathbb {R}^N) = |\varOmega \cap D| \qquad \text {and}\qquad \nu ^\infty (D \times \mathbb {S}^{N-1}) = \lambda _\nu (D) $$

for all Borel sets

\(D \subset \overline{\varOmega }\). So,

\(\nu \) can be understood as a classical Young measure with respect to

Open image in new window and target space

\(\mathbb {R}^N\) and

\(\nu ^\infty \) can be understood as a classical Young measure with respect to

\(\lambda _\nu \) and target space

\(\mathbb {S}^{N-1}\).

Alternative approaches to quantify concentration effects are varifolds [9–11, 125] and currents [115, 116, 134, 135]. Time-dependent generalized Young measures have also been developed for quasistatic evolution in plasticity theory [83–85]; in this context also see [95].

Lemma 12.14 and Lemma 12.28 are adapted from [6]. The characterization result of Theorem 12.19 also holds for BD-Young measures (i.e., those Young measures generated by symmetric derivatives of functions of bounded deformation), see [93].

The approach to \(\mathrm {BV}\)-lower semicontinuity through generalized Young measures in Theorem 12.25 was first implemented in this form in [228], the version we present here includes the shortening possible by the Kirchheim–Kristensen Theorem 10.13. We remark that it is also possible to incorporate *x*-dependence into the strategy we employed for the proof of Theorem 11.7, but the Young measure proof gives this result immediately (if the strong recession function of the integrand exists). We refer to [16] for weak* lower semicontinuity results for functionals defined on PDE-constrained measures.

We finally mention the very recent work [23], where the Souček space is considered as a more natural space of underlying deformations for BV-Young measures with boundary concentrations.