Journal of Convex Analysis 26 (2019), No. 2, 593--603
Copyright Heldermann Verlag 2019
A Fenchel-Moreau Theorem for L0-Valued Functions
Samuel Drapeau
Shanghai Jiao Tong University, School of Mathematical Sciences, and: China Academy of Financial Research, 211 West Huaihai Road, Shanghai, China
sdrapeau@saif.sjtu.edu.cn
Asgar Jamneshan
Dept. of Mathematics and Statistics, University of Konstanz, 78464 Konstanz, Germany
asgar.jamneshan@uni-konstanz.de
Michael Kupper
Dept. of Mathematics and Statistics, University of Konstanz, 78464 Konstanz, Germany
kupper@uni-konstanz.de
[Abstract-pdf]
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 \vspace*{-2mm} $$\smash{ f(x)=\sup_{y \in L^0(Y)} \left\{\langle x, y \rangle - f^\ast(y)\right\}, \quad x\in X,} $$ 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.
Keywords: Fenchel-Moreau theorem, vector duality, semi-continuous extension, conditional functional analysis.
MSC: 46A20, 03C90, 46B22
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