Projections on convex sets in the relaxed limit

The paper establishes the continuity of the best approximation, or the projection. of a function in L-p for p is an element of [1, infinity), on a closed convex set in the space, when the set varies and converges to a limit set in the Young-measure relaxation of the space. To this end a strong-type convergence and a convexity structure are identified on the space of Young measures. The appropriate convergence of sets with respect to which the continuity holds is the Mosco convergence of sets associated with the strong-type convergence of functions.