Pome

Corollary 13.9.445601E7EC1EE

Corollary 13.9.4.45601E7EC1EE Let $u_ k\in \mathrm{PSH}(X,\theta )$ satisfy $u_ k\leq \phi +C_ k$ for constants $C_ k$, let $u\in \mathcal E(X,\theta ,\phi )$, and suppose that $u_ k\to u$ in extended capacity. Then

\[ \langle (\theta +\mathrm{dd}^{\mathrm{c}}u_ k)^ n\rangle \longrightarrow \langle (\theta +\mathrm{dd}^{\mathrm{c}}u)^ n\rangle \]

weakly, and the masses converge. The same holds, without the constants $C_ k$, if $u\in \mathcal E(X,\theta )$; in particular it holds for every sequence in $\mathrm{PSH}(X,\theta )$ that decreases to an element of $\mathcal E(X,\theta )$.

Proof. By theorem 13.8.7, the mass of each $u_ k$ is at most that of $\phi $, which equals the mass of $u$. Hence theorem 13.7.5, with $p=n$, gives weak convergence, and its lower inequality together with this bound gives convergence of the masses. If $u\in \mathcal E(X,\theta )$, then by proposition 13.9.2 the mass of each $u_ k$ is at most $\operatorname {vol}(\{ \theta \} )$, which is the mass of $u$, and the same argument applies. Finally, decreasing sequences converge in extended capacity by lemma 13.7.3. $\square$

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