Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 26 (2019), No. 2, 593--603Copyright 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 [ Fulltext-pdf  (127  KB)] for subscribers only.