We introduce conditions on the construction of grand Lebesgue spaces on R-n which imply the validity of the Sobolev theorem for the Riesz fractional integrals I-alpha and the boundedness of the maximal operator, in such spaces. We also give an inversion of the operator I-alpha by means of hypersingular integrals, within the frameworks of the introduced spaces. We also proof the denseness of C-0(infinity)(R-n) in a subspace of the considered grand space.