A counterexample shows that not every locally $L^0$--convex topology is necessarily induced by a family of $L^0$--seminorms

Abstract

This paper constructs a counterexample showing that not every locally $L^0$--convex topology is necessarily induced by a family of $L^0$--seminorms. Random convex analysis is the analytic foundation for $L^0$--convex conditional risk measures, this counterexample, however, shows that a locally $L^0$--convex module is not a proper framework for random convex analysis. Further, this paper also gives a necessary and sufficient condition for a locally $L^0$--convex topology to be induced by a family of $L^0$--seminorms. Finally, we give some comments showing that based on random locally convex modules, we can establish a perfect random convex analysis to meet the needs of the study of $L^0$--convex conditional risk measures.

arXiv comment12 pages

References in corpus (4)

• Recent progress in random metric theory and its applications to conditional risk measures
• On random convex analysis -- the analytic foundation of the module approach to conditional risk measures
• On characterization of locally $L^0$-convex topologies induced by a family of $L^0$-seminorms
• A comprehensive connection between the basic results and properties derived from two kinds of topologies for a random locally convex module

Cited by in corpus (7)

• Versions of Eberlein-Šmulian and Amir-Lindenstrauss theorems in the framework of conditional sets
• Optimization of conditional convex risk measures
• The noncompact Schauder fixed point theorem in random normed modules and its applications
• $L^0$--convex compactness and its applications to random convex optimization and random variational inequalities
• On $d$--$\sigma$--stability in random metric spaces and its applications
• On random convex analysis
• A study of stability in locally $L^0$-convex modules and a conditional version of James' compactness theorem