Minimal topologies and $L_{p}$-spaces

Abstract

This paper deals with minimal topologies on Riesz spaces. A minimal topology is a Hausdorff locally solid topology that is coarser than any other Hausdorff locally solid topology on the space. It is shown that every minimal topology satisfies the Lebesgue property, that an Archimedean Riesz space can admit a locally convex-solid topology that is minimal if and only if the space is discrete, that $C[0,1]$ and $L([0,1])$ do not admit a minimal topology, and that the topology of convergence in measure on $Lp([0,1])(0 < p < \infty)$ is a minimal topology. A similar result is shown for certain Orlicz spaces.