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

On the Gamma convergence of functionals defined over pairs of measures and energy-measures