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.[1]
In probability theory, it means the standard probability space.[2]
The reason for the name abstract L-space is because every AL-space is isomorphic (as a Banach lattice) with some subspace of [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 (which is equal to X+) 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 𝜇 on K such that for every majorized and directed subset S of we have [1]