  A novel general framework for the study of $\Gamma$-convergence of
functionals defined over pairs of measures and energy-measures is introduced.
This theory allows us to identify the $\Gamma$-limit of these kind of
functionals by knowing the $\Gamma$-limit of the underlining energies. In
particular, the interaction between the functionals and the underlining
energies results, in the case these latter converge to a non continuous energy,
in an additional effect in the relaxation process. This study was motivated by
a question in the context of epitaxial growth evolution with adatoms.
Interesting cases of application of the general theory are also presented.
