Grand Lebesgue spaces with respect to measurable functions

Let 1<p<?. Given ??Rn a measurable set of finite Lebesgue measure, the norm of the grand Lebesgue spaces Lp)(?) is given by In this paper we consider the classical norm of the Grand Lebesgue space L^p) obtained considering a generic nonnegative measurable function ?(?). We find necessary and sufficient conditions on ? in order to get a functional equivalent to a Banach function norm, and we determine the "interesting" class Bp of functions ?, with the property that every generalized function norm is equivalent to a function norm built with ??Bp. We then define the Lp),?(?) spaces, prove some embedding results and conclude with the proof of the generalized Hardy inequality.