A Fenchel-Moreau Theorem for L̄ ⁰ -Valued Functions

We establish a Fenchel-Moreau type theorem for proper convex functions

, where

(X, Y, \langle \cdot,\cdot \rangle) ( X , Y , ⟨ ⋅ , ⋅ ⟩ )

is a dual pair of Banach spaces and

\bar L^0 L ˉ 0

is the space of all extended real-valued functions on a

\sigma 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 representation

\smash{ f(x)=\sup_{y \in L^0(Y)} \left\{\langle x, y \rangle - f^\ast(y)\right\}, \quad x\in X,} f ( x ) = sup ⁡ y ∈ L 0 ( Y ) { ⟨ x , y ⟩ − f ∗ ( y ) } , x ∈ X ,

where

L^0(Y) L 0 ( Y )

is the space of all strongly measurable functions with values in

Y Y

, and

\langle \cdot,\cdot \rangle ⟨ ⋅ , ⋅ ⟩

is understood pointwise almost everywhere. The proof is based on a conditional extension result and conditional functional analysis.