The forgotten parameter in grand lebesgue spaces

Let 1 < p < 8, e0 ?]0, p - 1], ? ? R-n be a Lebesgue measurable set of positive, finite measure, and let d : (0, p - 1] ? (0, 8) be such that We show that the linear set of functions d-(.) := d(.)f(Z f Lebesgue measurable on ? : sup d(e)- 0 1), which is equivalent to say that d ? ?2(0+). This means that in the case the parameter e0 plays a crucial role in the definition of a generalized grand Lebesgue space, namely, different values of e0 define different Banach function spaces. d -?/ ?2(0+),