}^{\ast }\left( \mathbb{Q}\right) =\sup_{X\in \mathfrak{M}_{b}\left( \Omega ,\mathcal{F}\right)
}\left\{ \int Z\left( -X\right) d\mathbb{P}-\rho \left( X\right) \right\},$$ where $Z:=d\mathbb{Q}/d\mathbb{P},$ we have that $\Psi \left( Z\right)
=\sup_{X\in \mathfrak{M}_{b}\left( \Omega ,\mathcal{F}\right) }\left\{ \Psi
_{2}^{X}\left( Z\right) \right\} $ is the supremum of a family of convex lower semicontinuous functions with respect to the topology $\sigma \left(
L^{1},L^{\infty }\right) $, and $\Psi \left( Z\right) $ preserves both properties.

$\left( b\right) $ For the Fenchel - Legendre transform (conjugate function) $\Psi ^{\ast }:\ L^{\infty }\left( \mathbb{P}\right) \longrightarrow \mathbb{R}$ for each $U\in L^{\infty }\left( \mathbb{P}\right) $$$\Psi ^{\ast }\left( U\right) =\sup\limits_{Z\in \delta \left( \mathcal{K}\right) }\left\{ \int ZUd\mathbb{P-}\Psi \left( Z\right) \right\}
=\sup\limits_{\mathbb{Q}\in \mathcal{Q}_{cont}}\left\{ \mathbb{E}_{\mathbb{Q}}\left[ U\right] \mathbb{-\psi }\left( \mathbb{Q}\right) \right\} \equiv
\rho \left( -U\right) .$$ From the lower semicontinuity of $\Psi $ w.r.t. the weak topology $\sigma \left( L^{1},L^{\infty }\right) $ that $\Psi =\Psi ^{\ast \ast }$. Considering the weak$^{\ast }$-topology $\sigma \left( L^{\infty }\left(
\mathbb{P}\right) ,L^{1}\left( \mathbb{P}\right) \right) $ for $Z=d\mathbb{Q}/d\mathbb{P}$ we have that $$\psi \left( \mathbb{Q}\right) =\Psi \left( Z\right) =\Psi ^{\ast \ast
}\left( Z\right) =\sup\limits_{U\in L^{\infty }\left( \mathbb{P}\right)
}\left\{ \int Z\left( -U\right) d\mathbb{P-}\Psi ^{\ast }\left( -U\right)
\right\} =\psi _{\rho }^{\ast }\left( \mathbb{Q}\right) .$$$\Box $

1.  As pointed out in Remark \[Remarkpsi(Q)=+oo\_for\_Q\_not\_&lt;&lt;\], we have that $$\mathbb{Q}\in \mathcal{Q}_{cont}\setminus \mathcal{Q}_{cont}^{\ll }\Longrightarrow \psi 