Corollary 19.2.13511230BE9AE5

Proof. Let $S^ j,S'^ j\in \mathcal{S}'(C)$ be elements such that $S^ j\to S$, $S'^ j\to S'$. Then it follows from 19.7.19 that $S^ j\cap S'^ j \to S$. Since $\operatorname{vol}$ is continuous, for large $j$, $S^ j\cap S'^ j$ has positive volume and hence lies in $\mathcal{S}'(C)$ by 19.7.25. We may therefore replace $S^ j$ by $S^ j\cap S'^ j$ and assume that $S^ j\subseteq S'^ j$. Hence, (19.2.14) follows from the continuity of $\Delta $ proved in theorem 19.7.32. $\square$