Internal characterization of Brezis -- Lieb spaces

Abstract

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 $p\in [1,\infty )$.

The first author would like to thank Organizers of the workshop Recent Advances in Banach lattices (18w5087) https://www.birs.ca/events/2018/5-day-workshops/18w5087 during which the work on the paper was significantly intensified