On compactness in $L^0$-modules

Abstract

Several results in functional analysis are extended to the setting of $L^0$-modules, where $L^0$ denotes the ring of all measurable functions $x:\Omega \to \mathbb{R}$. The focus is on results involving compactness. To this end, a notion of stable compactness is introduced, and it is argued that the conventional notion of compactness does not allow to establish a functional analytic discourse in $L^0$-modules. Several characterizations of stable compactness are discussed, and its importance in applications is highlighted.

arXiv comment25 pages

References in corpus (5)

• The algebra of conditional sets and the concepts of conditional topology and compactness
• Versions of Eberlein-Šmulian and Amir-Lindenstrauss theorems in the framework of conditional sets
• Sheaves and conditional sets
• Vector duality via conditional extension of dual pairs
• A Fenchel-Moreau theorem for ${\bar{L}}^0$-valued functions

Cited by in corpus (1)

• A perturbational duality approach in vector optimization