  In this paper, we continue to study random convex analysis. First, we
introduce the notion of an $L^0$--pre--barreled module. Then, we develop the
theory of random duality under the framework of a random locally convex module
endowed with the locally $L^0$--convex topology in order to establish a
characterization for a random locally convex module to be $L^0$--pre--barreled,
in particular we prove that the model space $L^{p}_{\mathcal{F}}(\mathcal{E})$
employed in the module approach to conditional risk measures is
$L^0$--pre--barreled, which forms the most difficult part of this paper.
Finally, we prove the continuity and subdifferentiability theorems for a proper
lower semicontinuous $L^0$--convex function on an $L^{0}$--pre--barreled random
locally convex module. So the principal results of this paper may be well
suited to the study of continuity and subdifferentiability for $L^0$--convex
conditional risk measures.
