Proposition 11.35.2.0C21983AC42D The functional $H_\varphi $ is convex and weakly lower semicontinuous. For every $F\in C(X)$,
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11.35.3
\begin{equation} B_\varphi (F) =\min _{\nu \in \mathcal P(X)} \left(H_\varphi (\nu )-\int _ XF\, \mathrm{d}\nu \right). \end{equation}
The minimizers are exactly $\mathcal A_\varphi (F)$.
Proof.
Each function of $\nu $ in the supremum defining $H_\varphi $ is affine and weakly continuous, so their supremum is convex and lower semicontinuous. By definition, $H_\varphi (\nu )-\int _ XF\, \mathrm{d}\nu \geq B_\varphi (F)$, with equality precisely when
\[ B_\varphi (G)+\int _ XG\, \mathrm{d}\nu \leq B_\varphi (F)+\int _ XF\, \mathrm{d}\nu \quad \text{for every }G\in C(X). \]
This is the defining inequality of $\mathcal A_\varphi (F)$, and the minimum is attained because $\mathcal A_\varphi (F)$ is nonempty by theorem 11.34.2.
$\square$
Pome
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