A Fenchel-Moreau Theorem for L0-Valued Functions

Authors

Samuel Drapeau, Asgar Jamneshan and Michael Kupper

Date

2019

Journal

Journal of Convex Analysis, 26(2):593-603

Abstract

In this article, we establish a Fenchel-Moreau type duality for proper convex and $\sigma(X,Y)$-lower semi-continuous functions $f\colon X\to \overline{L^0}$, where $(X,Y,\langle \cdot,\cdot \rangle)$ is a dual pair of Banach spaces and $\overline{L^0}$ is the collection of all equivalence classes of extended real-valued random variables. The duality result is based on several conditional extensions and analytic tools from conditional set theory. As an application of our main result we derive a vector duality for increasing convex functions from a Banach lattice $X$ into $L^0$ in terms of dual elements in the Bochner-Lebesgue space $L^0(X^\ast)$. For convex functions $f\colon L^\infty \to L^0$ satisfying the Fatou continuity property, we derive a vector duality in terms of dual elements in the Bochner-Lebesgue space $L^0(L^1)$.

Keywords

Vector duality, conditional extension of dual pairs, conditional set theory, Bochner-Lebesgue spaces, vector optimization, vector-valued risk measures

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