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In this paper, we continue to study random convex analysis. First, weintroduce the notion of an $L^0$--pre--barreled module. Then, we develop thetheory of random duality under the framework of a random locally convex moduleendowed with the locally $L^0$--convex topology in order to establish acharacterization 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 properlower semicontinuous $L^0$--convex function on an $L^{0}$--pre--barreled randomlocally convex module. So the principal results of this paper may be wellsuited to the study of continuity and subdifferentiability for $L^0$--convexconditional risk measures.

Random convex analysis (II): continuity and subdifferentiability theorems in $L^{0}$--pre--barreled random locally convex modules