risk is concentrated on the set $\mathcal{Q}$. Later, Krätschmer [@Kraetschmer; @2005 Prop. 3 p. 601] established that the sequential continuity from below is not only a sufficient but also a necessary condition in order to have a representation, by means of the minimal penalty function in terms of probability measures.

We denote by $\mathcal{Q}_{\ll }(\mathbb{P})$ the subclass of absolutely continuous probability measure with respect to $\mathbb{P}$ and by $\mathcal{Q}_{\approx }\left( \mathbb{P}\right) $ the subclass of equivalent probability measure. Of course, $\mathcal{Q}_{\approx }\left( \mathbb{P}\right) \subset \mathcal{Q}_{\ll }(\mathbb{P})\subset \mathcal{Q}\left(
\Omega ,\mathcal{F}\right) $.

\[Remarkpsi(Q)=+oo\_for\_Q\_not\_&lt;&lt;\] When a convex risk measures in $\mathcal{X}:=L^{\infty }\left( \mathbb{P}\right) $ satisfies the property $$\rho \left( X\right) =\rho \left( Y\right) \text{ if }X=Y\ \mathbb{P}\text{-a.s.}  \label{rho(X)=rho(Y)_for_X=Y}$$and is represented by a penalty function $\psi $ as in $\left( \ref{Static_CMR_induced_by_phi}\right) $, we have that $$\mathbb{Q}\in \mathcal{Q}_{cont}\setminus \mathcal{Q}_{cont}^{\ll
}\Longrightarrow \psi \left( \mathbb{Q}\right) =+\infty ,
\label{psi(Q)=+oo_for_Q_not_<<}$$where $\mathcal{Q}_{cont}^{\ll }$ is the set of contents absolutely continuous with respect to $\mathbb{P}$; see [FoellSch 2004]{}.

Minimal penalty functions [Subsect:\_Minimal\_penalty\_functions]{}
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The minimality property of the penalty function turns out to be quite relevant, and it is a desirable property that is not easy to prove in general. For instance, in the study of robust portfolio optimization problems (see, 