The supporting probabilities of continuous approximations of a bounded measurable logarithmic density lie in one sublevel set of $H_\varphi $. Uniform integrability then allows both the density and the supporting measure to vary when we pass to the limit in the supporting inequality.
Theorem 11.36.1.99653D7759E2 For every $F\in L^\infty (\mathrm{d}V)$ there is a nonempty set of absolutely continuous probability measures, again denoted $\mathcal A_\varphi (F)$, characterized by
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11.36.2
\begin{equation} B_\varphi (G)\leq B_\varphi (F)-\int _ X(G-F)\, \mathrm{d}\nu \qquad \text{for every }G\in L^\infty (\mathrm{d}V). \end{equation}
For continuous $F$, this set agrees with the one defined by (11.34.1). Formula (11.35.3) also holds for bounded measurable $F$, with these measures as its minimizers; there the objective is interpreted as $+\infty $ whenever $H_\varphi (\nu )=+\infty $, and otherwise $\nu $ is absolutely continuous, so that the integral of the $L^\infty $ class $F$ is well defined.
Suppose that $F_ j,F\in L^\infty (\mathrm{d}V)$ are uniformly bounded and $F_ j\to F$ in $L^1(\mathrm{d}V)$, and let $\nu _ j\in \mathcal A_\varphi (F_ j)$. Then the densities of the $\nu _ j$ are uniformly integrable, and every weak $L^1$ cluster point belongs to $\mathcal A_\varphi (F)$. If the latter is a singleton, the whole sequence converges to it weakly in $L^1(\mathrm{d}V)$.
Proof.
First, if $H_\varphi (\nu )<\infty $, the measure $\nu $ is absolutely continuous by theorem 11.35.4. A bounded measurable $G$ admits uniformly bounded continuous approximations $G_ k\to G$ in $L^1(\mathrm{d}V)$, and absolute continuity together with the common bound gives $\int _ XG_ k\, \mathrm{d}\nu \to \int _ XG\, \mathrm{d}\nu $. Combined with theorem 11.33.5, this proves
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11.36.3
\begin{equation} H_\varphi (\nu ) =\sup _{G\in L^\infty (\mathrm{d}V)} \left(B_\varphi (G)+\int _ XG\, \mathrm{d}\nu \right) \end{equation}
for such $\nu $.
To construct supporting probabilities at $F$, choose continuous $F_ j$ with $|F_ j|\leq M=\| F\| _\infty $ and $F_ j\to F$ in $L^1(\mathrm{d}V)$, and choose $\nu _ j\in \mathcal A_\varphi (F_ j)$. The minimum formula of proposition 11.35.2 and the Lipschitz bound give
\[ H_\varphi (\nu _ j) =B_\varphi (F_ j)+\int _ XF_ j\, \mathrm{d}\nu _ j \leq B_\varphi (0)+2M. \]
By theorem 11.35.4, a subsequence converges weakly in $L^1$ to a probability measure $\nu $. Uniform absolute continuity gives
\[ \int _ X|F_ j-F|\, \mathrm{d}\nu _ j\longrightarrow 0, \]
by splitting again at a fixed small positive threshold: the sets where $|F_ j-F|$ exceeds it have $\mathrm{d}V$-measure tending to zero. Since $F$ is bounded and measurable, weak $L^1$ convergence applies to its integral. Lower semicontinuity of $H_\varphi $ and continuity of $B_\varphi $ yield
\[ H_\varphi (\nu ) \leq \liminf _ j H_\varphi (\nu _ j) =B_\varphi (F)+\int _ XF\, \mathrm{d}\nu . \]
The reverse inequality follows from (11.36.3) with $G=F$. Equality, combined with the same supremum formula, proves (11.36.2) for $\nu $ and the minimum formula.
For continuous $F$, every measure satisfying (11.34.1) has finite dual value by proposition 11.35.2, so (11.36.3) extends its supporting inequality to all bounded measurable tests; the converse inclusion is trivial. Thus the two definitions agree. Finally, an absolutely continuous probability satisfying (11.36.2) has finite dual value and attains the minimum, so the characterization is exact.
For the stability assertion, we start from the given $F_ j$ and $\nu _ j$ instead of an approximating construction. The common uniform bound again places all $\nu _ j$ in a single finite sublevel set of $H_\varphi $, and the same compactness and limiting argument shows that every weak $L^1$ cluster point attains the minimum for $F$. If the set of minimizers is a singleton, compactness forces the whole sequence to converge to its member.
$\square$
Pome
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