  A Fenchel-Moreau type duality for proper convex and lower semi-continuous
functions $f\colon X\to \overline{L^0}$ is established where $(X,Y,\langle
\cdot,\cdot \rangle)$ is a dual pair of Banach spaces and $\overline{L^0}$ is
the set of all extended real-valued measurable functions. We provide a concept
of lower semi-continuity which is shown to be equivalent to the existence of a
dual representation in terms of elements in the Bochner space $L^0(Y)$. To
derive the duality result, several conditional completions and extensions are
constructed.
  This is an earlier version of arXiv e-print 1708.03127, where the main
results were formulated in an abstract setting of conditional completions,
conditional extensions and conditional real numbers.
