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To provide a solid analytic foundation for the module approach to conditionalrisk measures, this paper establishes a complete random convex analysis overrandom locally convex modules by simultaneously considering the two kinds oftopologies (namely the $(\varepsilon,\lambda)$--topology and the locally$L^0$-- convex topology). Then, we make use of the advantage of the$(\varepsilon,\lambda)$--topology and grasp the local property of $L^0$--convexconditional risk measures to prove that every $L^{0}$--convex$L^{p}$--conditional risk measure ($1\leq p\leq+\infty$) can be uniquelyextended to an $L^{0}$--convex $L^{p}_{\mathcal{F}}(\mathcal{E})$--conditionalrisk measure and that the dual representation theorem of the former can also beregarded as a special case of that of the latter, which shows that the study of$L^p$--conditional risk measures can be incorporated into that of$L^{p}_{\mathcal{F}}(\mathcal{E})$--conditional risk measures. In particular,in the process we find that combining the countable concatenation hull of a setand the local property of conditional risk measures is a very useful analyticskill that may considerably simplify and improve the study of $L^{0}$--convexconditional risk measures.

On random convex analysis -- the analytic foundation of the module approach to conditional risk measures