  Brezis-Lieb lemma is a refinement of Fatou lemma providing an evaluation of
the gap between the integral for a sequence and the integral for its pointwise
limit. This note studies the question if such gap can be evaluated when there
is no a.e. convergence. In particular, it gives the same lower bound for the
gap in L^p as the gap in the Brezis-Lieb lemma (including the case
vector-valued functions) provided that p is greater or equal than 3 and the
sequence converges both weakly and weakly in the sense of a duality map. It
also shows that the statement is false if p<3. An application is given in form
of a Brezis-Lieb lemma for gradients.
