
Journal of Convex Analysis 26 (2019), No. 2, 593603 Copyright Heldermann Verlag 2019 A FenchelMoreau Theorem for L^{0}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@unikonstanz.de Michael Kupper Dept. of Mathematics and Statistics, University of Konstanz, 78464 Konstanz, Germany kupper@unikonstanz.de [Abstractpdf] We establish a FenchelMoreau 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 realvalued functions on a $\sigma$finite measure space. We introduce the concept of stable lower semicontinuity 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: FenchelMoreau theorem, vector duality, semicontinuous extension, conditional functional analysis. MSC: 46A20, 03C90, 46B22 [ Fulltextpdf (127 KB)] for subscribers only. 