  In order to find an extension of Brezis -- Lieb's lemma to the case of nets,
we replace the almost everywhere convergence by the unbounded order convergence
and introduce the Brezis -- Lieb property in normed lattices. Then we identify
a wide class of Banach lattices in which the Brezis -- Lieb lemma holds true.
Among other things, it gives an extension of the Brezis -- Lieb lemma for nets
in $L^p$ for $p\in [1,\infty)$.
