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 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 [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]

See also


  1. ^ a b c d e f g Schaefer & Wolff 1999, pp. 242–250.
  2. ^ Takeyuki Hida, Stationary Stochastic Processes, p. 21