A Fenchel-Moreau Theorem for (L)over-bar0-Valued Functions

We establish a Fenchel-Moreau type theorem for proper convex functions f : X -> (L) over bar (0), where (X,Y,<.,.>) is a dual pair of Banach spaces and (L) over bar (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 f(x) = sup(y is an element of L0(Y)) {< x, y > - f*(y)}, x is an element of X, where L-0(Y) is the space of all strongly measurable functions with values in Y, and <.,.> is understood pointwise almost everywhere. The proof is based on a conditional extension result and conditional functional analysis.