  This paper constructs a counterexample showing that not every locally
$L^0$--convex topology is necessarily induced by a family of $L^0$--seminorms.
Random convex analysis is the analytic foundation for $L^0$--convex conditional
risk measures, this counterexample, however, shows that a locally $L^0$--convex
module is not a proper framework for random convex analysis. Further, this
paper also gives a necessary and sufficient condition for a locally
$L^0$--convex topology to be induced by a family of $L^0$--seminorms. Finally,
we give some comments showing that based on random locally convex modules, we
can establish a perfect random convex analysis to meet the needs of the study
of $L^0$--convex conditional risk measures.
