The space of scalarly integrable functions for a Fréchet-space-valued measure

Abstract

The space L1 w (ν) of all scalarly integrable functions with respect to a Fréchet-space-valued vector measure ν is shown to be a complete Fréchet lattice with the σ-Fatou property which contains the (traditional) space L1(ν), of all ν-integrable functions. Indeed, L1(ν) is the σ-order continuous part of L1 w (ν). Every Fréchet lattice with the σ-Fatou property and containing a weak unit in its σ-order continuous part is Fréchet lattice isomorphic to a space of the kind L1 w (ν).