  The framework of this paper is that of risk measuring under uncertainty,
which is when no reference probability measure is given. To every regular
convex risk measure on ${\cal C}_b(\Omega)$, we associate a unique equivalence
class of probability measures on Borel sets, characterizing the riskless non
positive elements of ${\cal C}_b(\Omega)$. We prove that the convex risk
measure has a dual representation with a countable set of probability measures
absolutely continuous with respect to a certain probability measure in this
class.
  To get these results we study the topological properties of the dual of the
Banach space $L^1(c)$ associated to a capacity $c$.
  As application we obtain that every $G$-expectation $\E$ has a representation
with a countable set of probability measures absolutely continuous with respect
to a probability measure $P$ such that $P(|f|)=0$ iff $\E(|f|)=0$. We also
apply our results to the case of uncertain volatility.

Risk Management;Probability