A proof of the Ghoussoub-Preiss theorem by the ε-perturbation of Brezis-Nirenberg

In this note, a proof of the Ghoussoub-Preiss theorem is presented by using the ε−perturbation as introduced by Brezis-Nirenberg. Thus, besides the deformation lemma, other advanced tools such as the Radon measures space, sub-differential, or the theory of non-differentiable functions, are avoided. Our new argument is a lemma of local type which is used in combination with other main ingredients like the Ekeland variational principle and the pseudo-gradient lemma, for which a new proof is proposed as a consequence of the Michael selection theorem.