  A perturbational vector duality approach for objective functions $f\colon
X\to \bar{L}^0$ is developed, where $X$ is a Banach space and $\bar{L}^0$ is
the space of extended real valued functions on a measure space, which extends
the perturbational approach from the scalar case. The corresponding strong
duality statement is proved under a closedness type regularity condition.
Optimality conditions and a Moreau-Rockafellar type formula are provided. The
results are specialized for constrained and unconstrained problems. Examples of
integral operators and risk measures are discussed.
