  We establish a Fenchel-Moreau 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 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 $$\smash{ f(x)=\sup_{y \in L^0(Y)} \left\{\langle x, y
\rangle - f^\ast(y)\right\}, \quad x\in X,} $$ 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.
