Ekeland's Variational Principle for An $\bar{L}^{0}-$Valued Function on A Complete Random Metric Space

TL;DR

This paper extends Ekeland's variational principle to $ar{L}^{0}$-valued functions on complete random metric spaces, with applications to the Bishop-Phelps theorem in random normed modules, motivated by conditional risk measures.

Contribution

It develops a general form of Ekeland's variational principle for $ar{L}^{0}$-valued functions on complete random metric spaces and applies it to establish the Bishop-Phelps theorem in this context.

Findings

Proved a general Ekeland's variational principle for $ar{L}^{0}$-valued functions.

Derived a refined version for local functions on random normed modules.

Established the Bishop-Phelps theorem within the framework of random conjugate spaces.

Abstract

Motivated by the recent work on conditional risk measures, this paper studies the Ekeland's variational principle for a proper, lower semicontinuous and lower bounded $\bar{L}^{0}-$valued function, where $\bar{L}^{0}$ is the set of equivalence classes of extended real-valued random variables on a probability space. First, we prove a general form of Ekeland's variational principle for such a function defined on a complete random metric space. Then, we give a more precise form of Ekeland's variational principle for such a local function on a complete random normed module. Finally, as applications, we establish the Bishop-Phelps theorem in a complete random normed module under the framework of random conjugate spaces.