A study of stability in locally $L^0$-convex modules and a conditional version of James' compactness theorem

TL;DR

This paper explores the structure of locally $L^0$-convex modules, characterizes which are equivalent to locally convex vector spaces in conditional set theory, and extends classical theorems like James' compactness theorem to the conditional setting.

Contribution

It identifies subclasses of locally $L^0$-convex modules that correspond to locally convex vector spaces and develops conditional versions of key theorems in functional analysis.

Findings

Characterization of subclasses of locally $L^0$-convex modules

Conditional James' theorem for weak compactness

Conditional Fatou and Lebesgue properties for risk measures

Abstract

Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N. Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct. Anal. 256(12), 3996-4029 (2009)] as the analytic basis for the study of conditional risk measures. Later, the algebra of conditional sets was introduced in [S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper. The algebra of conditional sets and the concepts of conditional topology and compactness. J. Math. Anal. Appl. 437(1), 561-589 (2016)]. In this paper we study locally $L^0$-convex modules, and find exactly which subclass of locally $L^0$-convex modules can be identified with the class of locally convex vector spaces within the context of conditional set theory. Second, we provide a version of the classical James' theorem of characterization of weak compactness for conditional Banach spaces. Finally, we state a conditional version of the Fatou and Lebesgue properties for conditional convex risk measures and, as application of the developed theory, we stablish a version of the so-called Jouini-Schachermayer-Touzi theorem for robust representation of conditional convex risk measures defined on a $L^\infty$-type module.