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 $$(X, \| \cdot \|)$$ 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 (as a Banach lattice) with some subspace of $$L^1(\mu).$$ 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 $$X^{\prime}$$ (which is equal to X+) of an AL-space X is a Banach lattice that can be identified with $$C_{\R} ( K )$$, 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 𝜇 on K such that for every majorized and directed subset S of $$C_{\R} ( K ),$$ we have $$\lim_{f \in S} \mu ( f ) = \mu ( \sup S ).$$