$ is the coarsest topology on $L^{1}\left( \mathbb{P}\right) $ under which every linear operator is continuous, and hence $\Psi _{0}^{X}\left( Z\right) :=\mathbb{E}_{\mathbb{P}}\left[ Z\left(
-X\right) \right] $, with $Z\in L^1$, is a continuous function for each $X\in
\mathfrak{M}_{b}\left( \Omega ,\mathcal{F}\right) $ fixed. For $\delta \left(
\mathcal{K}\right) :=\left\{ Z:Z=d\mathbb{Q}/d\mathbb{P}\text{ with }\mathbb{Q}\in \mathcal{K}\right\} $ we have that$$\Psi _{1}^{X}\left( Z\right) :=\Psi _{0}^{X}\left( Z\right) \mathbf{1}_{\delta \left( \mathcal{K}\right) }\left( Z\right) +\infty \times \mathbf{1}_{L^{1}\setminus \delta \left( \mathcal{K}\right) }\left( Z\right)$$is clearly lower semicontinuous on $\delta \left( \mathcal{K}\right) .$ For $Z^{\prime }\in L^{1}\left( \mathbb{P}\right) \setminus \delta \left(
\mathcal{K}\right) $ arbitrary fixed we have from Hahn-Banach’s Theorem that there is a continuous lineal functional $l\left( Z\right) $ with $l\left(
Z^{\prime }\right) <\inf_{Z\in \delta \left( \mathcal{K}\right) }l\left(
Z\right) $. Taking $\varepsilon :=\frac{1}{2}\left\{ \inf_{Z\in \delta
\left( \mathcal{K}\right) }l\left( Z\right) -l\left( Z^{\prime }\right)
\right\} $ we have that the weak open ball $B\left( Z^{\prime },\varepsilon
\right) :=\left\{ Z\in L^{1}\left( \mathbb{P}\right) :\left\vert l\left(
Z^{\prime }\right) -l\left( Z\right) \right\vert <\varepsilon \right\} $ satisfies $B\left( Z^{\prime },\varepsilon \right) \cap \delta \left( \mathcal{K}\right) =\varnothing .$ Therefore, $\Psi _{1}^{X}\left( Z\right) $ is weak lower semicontinuous on $L^{1}\left( \mathbb{P}\right) ,$ as well as $\Psi
_{2}^{X}\left( Z\right) :=\Psi _{1}^{X}\left( Z\right) -\rho \left( X\right)
.$ If $$\psi \left( \mathbb{Q}\right) =\psi _{\rho 