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

The space L w 1 ( 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 L 1 ( ν ) , of all ν-integrable functions. Indeed, L 1 ( ν ) is the σ-order continuous part of L w 1 ( 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 L w 1 ( ν ) .