  The purpose of this paper is to give a selective survey on recent progress in
random metric theory and its applications to conditional risk measures. This
paper includes eight sections. Section 1 is a longer introduction, which gives
a brief introduction to random metric theory, risk measures and conditional
risk measures. Section 2 gives the central framework in random metric theory,
topological structures, important examples, the notions of a random conjugate
space and the Hahn-Banach theorems for random linear functionals. Section 3
gives several important representation theorems for random conjugate spaces.
Section 4 gives characterizations for a complete random normed module to be
random reflexive. Section 5 gives hyperplane separation theorems currently
available in random locally convex modules. Section 6 gives the theory of
random duality with respect to the locally $L^{0}-$convex topology and in
particular a characterization for a locally $L^{0}-$convex module to be
$L^{0}-$pre$-$barreled. Section 7 gives some basic results on $L^{0}-$convex
analysis together with some applications to conditional risk measures. Finally,
Section 8 is devoted to extensions of conditional convex risk measures, which
shows that every representable $L^{\infty}-$type of conditional convex risk
measure and every continuous $L^{p}-$type of convex conditional risk measure
($1\leq p<+\infty$) can be extended to an $L^{\infty}_{\cal F}({\cal E})-$type
of $\sigma_{\epsilon,\lambda}(L^{\infty}_{\cal F}({\cal E}), L^{1}_{\cal
F}({\cal E}))-$lower semicontinuous conditional convex risk measure and an
$L^{p}_{\cal F}({\cal E})-$type of ${\cal T}_{\epsilon,\lambda}-$continuous
conditional convex risk measure ($1\leq p<+\infty$), respectively.
