This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (uj , Duj )j for (uj )j ⊂ BV(Ω; R m ) under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functionalto the space BV(Ω; R m ). Lower semicontinuity results of this type were first obtained by Fonseca & Müller [Arch. Ration. Mech. Anal. 123 (1993), 1-49] and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that f be Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising F in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.