Topological Properties of the Complex Vector Lattice of Bounded Measurable Functions

Let Σ be a σ-algebra of subsets of a nonempty set ？. Let be the complex vector lattice of bounded Σ-measurable complex-valued functions on ？ and let be the Banach space of all bounded countably additive complex-valued measures on ？. We study locally solid topologies on . In particular, it is shown that the Mackey topology is the finest locally convex-solid σ-Lebesgue topology on .