  This paper is devoted to the construction of generalized multi-scale Young
measures, which are the extension of Pedregal's multi-scale Young measures
[Trans. Amer. Math. Soc. 358 (2006), pp. 591-602] to the setting of generalized
Young measures introduced by DiPerna and Majda [Comm. Math. Phys. 108 (1987),
pp. 667-689]. As a tool for variational problems, these are well-suited objects
for the study (at different length-scales) of oscillation and concentration
effects of convergent sequences of measures. Important properties of
multi-scale Young measures such as compactness, representation of non-linear
compositions, localization principles, and differential constraints are
extensively developed in the second part of this paper. As an application, we
use this framework to address the $\Gamma$-limit characterization of the
homogenized limit of convex integrals defined on spaces of measures satisfying
a general linear PDE constraint.
