Internal characterization of Brezis -- Lieb spaces

TL;DR

This paper extends the Brezis--Lieb lemma to nets in Banach lattices by replacing almost everywhere convergence with unbounded order convergence, broadening its applicability.

Contribution

It introduces the Brezis--Lieb property in normed lattices and identifies classes of Banach lattices where the lemma holds for nets.

Findings

Extension of Brezis--Lieb lemma to nets in $L^p$ spaces.

Identification of Banach lattices satisfying the Brezis--Lieb property.

Generalization of convergence conditions in lattice theory.

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