A Fenchel-Moreau theorem for $\bar L^0$-valued functions

TL;DR

This paper extends the Fenchel-Moreau duality theorem to convex functions valued in the space of extended real-valued functions on a measure space, introducing stable lower semi-continuity for dual representation.

Contribution

It establishes a Fenchel-Moreau type duality theorem for $ar L^0$-valued convex functions using conditional analysis and introduces the concept of stable lower semi-continuity.

Findings

Proves a dual representation for convex functions in $ar L^0$.

Defines and characterizes stable lower semi-continuity.

Uses conditional functional analysis techniques.

Abstract

We establish a Fenchel-Moreau type theorem for proper convex functions $f\colon X\to \bar{L}^0$, where $(X, Y, \langle \cdot,\cdot \rangle)$ is a dual pair of Banach spaces and $\bar L^0$ is the space of all extended real-valued functions on a $\sigma$-finite measure space. We introduce the concept of stable lower semi-continuity which is shown to be equivalent to the existence of a dual representation $$\smash{ f(x)=\sup_{y \in L^0(Y)} \left\{\langle x, y \rangle - f^\ast(y)\right\}, \quad x\in X,} $$ where $L^0(Y)$ is the space of all strongly measurable functions with values in $Y$, and $\langle \cdot,\cdot \rangle$ is understood pointwise almost everywhere. The proof is based on a conditional extension result and conditional functional analysis.