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

Abstract

We establish a Fenchel-Moreau theorem for proper convex functions f : X → L0 , where (X, Y, •, • ) is a dual pair of Banach spaces and L0 is the space of all extended real-valued functions on a σ-finite measure space. We introduce the concept of stable lower semi-continuity which is shown to be equivalent to the existence of a dual representationwhere L 0 (Y ) is the space of all strongly measurable functions with values in Y , and•, • is understood pointwise almost everywhere.

Measures and Integrals in Conditional Set Theory

“…See [6,11,12,15,20,23] for further results in conditional analysis and conditional set theory. For applications of conditional analysis, we refer to [2,3,5,7,8,10,13,14,17,19]. See [4,16] for other related results.…”

On compactness in $L^0$-modules

“…In [34] the connection between vector spaces in a Boolean-valued model and modules in the standard model are investigated systematically. Conditional analysis is applied to stochastic control problems, dynamic risk sharing, representation of preferences, duality of conditional risk measures, equilibrium pricing, Principal-Agent problems and vector duality in [2,4,5,6,8,9,12,15,16,31,38,39].…”

“…Here we understand convergence as convergence of (xn) as a net 12. A stable subsequence (yn) is said to be a stable subsequence of (xn) if there exists an increasing stable sequence (n k ) with y k = xn k for all k ∈ L 0 s (N).…”