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 ${\displaystyle (X,\|\cdot \|)}$ whose norm is additive on the positive cone of X.¹

In probability theory, it means the standard probability space.²

The strong dual of an AM-space with unit is an AL-space.¹

The reason for the name abstract L-space is because every AL-space is isomorphic (as a Banach lattice) with some subspace of ${\displaystyle L^{1}(\mu ).}$¹ 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.¹ Each order interval in an AL-space is weakly compact.¹

The strong dual of an AL-space is an AM-space with unit.¹ The continuous dual space ${\displaystyle X^{\prime }}$ (which is equal to X⁺) of an AL-space X is a Banach lattice that can be identified with ${\displaystyle C_{\mathbb {R} }(K)}$, 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 ${\displaystyle C_{\mathbb {R} }(K),}$ we have ${\displaystyle \lim _{f\in S}\mu (f)=\mu (\sup S).}$¹