In the present paper, the authors have established necessary and sufficient conditions ensuring that a given DiPerna-Majda measure is generated by gradients without any restrictions on the generating sequence, which extends a result of A. Kałamajska and M. Kružík [ESAIM, Control Optim. Calc. Var. 14, No. 1, 71–104 (2008; Zbl 1140.49009)] (here the full characterization was possible only for sequences subject to a fixed Dirichlet boundary condition). As an application, a relaxation result for non-coercive multiple-integral functionals is proved.