On characterization of locally $L^0$-convex topologies induced by a   family of $L^0$-seminorms
  The purpose of this paper is to provide a characterization of the topological $L^0$-modules whose topology is induced by a family of $L^0$-seminorms using the gauge function for $L^0$-modules. Taking advantage of these ideas we will give a counterexample of a locally $L^0$-convex module whose topology is not induced by a family of $L^0$-seminorms. 
Jos\'e Miguel Zapata Garc\'ia
math.FA