  To provide a solid analytic foundation for the module approach to conditional
risk measures, this paper establishes a complete random convex analysis over
random locally convex modules by simultaneously considering the two kinds of
topologies (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 $L^0$--convex
conditional risk measures to prove that every $L^{0}$--convex
$L^{p}$--conditional risk measure ($1\leq p\leq+\infty$) can be uniquely
extended to an $L^{0}$--convex $L^{p}_{\mathcal{F}}(\mathcal{E})$--conditional
risk measure and that the dual representation theorem of the former can also be
regarded 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 set
and the local property of conditional risk measures is a very useful analytic
skill that may considerably simplify and improve the study of $L^{0}$--convex
conditional risk measures.

Functional Analysis;Risk Management