Lemma 8.11.11.978D54F1054C Fix $0\leq s\leq m$ and the outer smooth capacity $c_ s^*$ on $U$. Let signed Radon measures $\nu _ j\to \nu $ weakly have uniformly bounded local variations, with all variations dominated on compact subsets by $C c_ s^*$. Suppose, for $1\leq i\leq \ell $, that bounded Borel functions $b_ i^ j\downarrow b_ i$ have one common bound and every member of these countable families is locally quasicontinuous for this capacity. For any continuous real $F$ on a compact box containing their ranges,
\[ F(b_1^ j,\ldots ,b_\ell ^ j)\nu _ j \longrightarrow F(b_1,\ldots ,b_\ell )\nu \]
weakly as signed Radon measures.
Proof.
Fix $\phi \in C_ c(U)$ and a compact $K\Subset U$ containing its support. Allocate summable exceptional capacities to all the functions. Their open union $G$ has $c_ s(G)<\epsilon $, and every function is continuous on the compact set $A=K\setminus G$. Dini's theorem gives $b_ i^ j\to b_ i$ uniformly on $A$ for each $i$. Uniform continuity of $F$ on the compact box therefore gives $f_ j\to f$ uniformly on $A$, where $f_ j=F(b_1^ j,\ldots ,b_\ell ^ j)$ and $f=F(b_1,\ldots ,b_\ell )$. Put $M=\| F\| _\infty $. Extend $f|_ A$ to a continuous $g$ on $U$ with $|g|\leq M$; if $A$ is empty, take $g=0$. Then
\[ \begin{aligned} \left|\int \phi f_ j\, \mathrm d\nu _ j- \int \phi f\, \mathrm d\nu \right| & \leq \| \phi \| _\infty \| f_ j-f\| _ A|\nu _ j|(K) +\left|\int \phi g\, \mathrm d(\nu _ j-\nu )\right|\\ & \quad +2M\| \phi \| _\infty \bigl (|\nu _ j|(K\cap G)+|\nu |(K\cap G)\bigr ). \end{aligned} \]
For fixed $G$, the first two terms tend to zero. The last term is at most $4MC\| \phi \| _\infty \epsilon $. Let $\epsilon $ tend to zero. Weak convergence is applied to $\phi g$ on the whole measure.
$\square$
Pome
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