Functional representation of vector lattices

Abstract:

Every vector lattice is represented in the lattice of distribution functions valued in the complete Boolean algebra of its annihilators; the representation is complete join and positive multiple preserving and subadditive; restricted to the solid vector sublattice without infinitesimals, it preserves the full structure (including any existing infinite lattice extrema) and is faithful. Identifying the distribution with the continuous real-valued functions on the extremally disconnected Stone space of the algebra yields a representation which, specialized to Archimedean vector lattices, embeds them in the densely finite-valued continuous functions; identifying with the equivalence classes of functions measurable for a $\sigma$-field modulo a $\sigma$-ideal of "null sets" yields a representation which, specialized to Archimedean vector lattices, embeds them in the classes of a.e. finite functions. This is used to give simple proofs of Freudenthal’s spectral theorem and Kakutani’s structure theorem for $L$-spaces.