Topological Properties of the Complex Vector Lattice of Bounded Measurable Functions

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