Abstract L-space

In mathematics, specifically in order theory and functional analysis, an abstract L-space, an AL-space, or an abstract Lebesgue space is a Banach lattice whose norm is additive on the positive cone of X.
In probability theory, it means the standard probability space.

Examples

The strong dual of an AM-space with unit is an AL-space.

Properties

The reason for the name abstract L-space is because every AL-space is isomorphic with some subspace of
Every AL-space X is an order complete vector lattice of minimal type;
however, the order dual of X, denoted by X⁺, is not of minimal type unless X is finite-dimensional.
Each order interval in an AL-space is weakly compact.
The strong dual of an AL-space is an AM-space with unit.
The continuous dual space of an AL-space X is a Banach lattice that can be identified with, where K is a compact extremally disconnected topological space;
furthermore, under the evaluation map, X is isomorphic with the band of all real Radon measures ? on K such that for every majorized and directed subset S of we have