Relaxation and gamma-convergence of supremal functionals

We prove that the Gamma-limit in L^1_μ of a sequence of supremal functionals of the form F_k(u) = μ-ess sup f_k(x, u) is itself a supremal functional. We show by a counterexample that, in general, the function which represents the Gamma- lim F(·,B) of a sequence of functionals F_k(u,B) = μ-ess sup_B f_k(x, u) can depend on the set B and we give a necessary and sufficient condition to represent F in the supremal form F(u,B) = μ-ess sup_B f(x, u). As a corollary, if f represents a supremal functional, then the level convex envelope of f represents its weak* lower semicontinuous envelope.