Proof.
Since $d_ S$ is a pseudometric, in order to establish the continuity of mixed masses, it suffices to consider sequences instead of nets. So we may assume that $I=\mathbb {Z}_{>0}$ as ordered sets.
Step 1. We reduce to the case where $\varphi _ j^ k,\varphi _ j$ all have positive masses and there is a constant $\delta >0$ such that for all $j=1,\ldots ,n$ and $k\in I$,
\[ \int _ X \theta _{j,\varphi _ j^ k}^ n>\delta . \]
Assuming in this case, the theorem has been proved, let us handle the general case. Take $t\in (0,1)$. By 16.9.27, we have
\[ (1-t)\varphi _ j^ k+tV_{\theta _ j}\xrightarrow {d_ S}(1-t)\varphi _ j+tV_{\theta _ j} \]
as $k\to \infty $ for each $j=1,\ldots ,n$. Note that for all $k\in I$,
\[ \int _ X \theta ^ n_{j,(1-t)\varphi _ j^ k+tV_{\theta _ j}}\geq t^ n\int _ X \theta _{V_{\theta _ j}}^ n>0. \]
Assume that we have proved the special case of the theorem, we have
\[ \begin{split} & \lim _{k\to \infty }\int _ X\theta _{1,(1-t)\varphi _1^ k+tV_{\theta _1}}\wedge \cdots \wedge \theta _{n,(1-t)\varphi _ n^ k+tV_{\theta _ n}}\\ =& \int _ X\theta _{1,(1-t)\varphi _1+tV_{\theta _1}}\wedge \cdots \wedge \theta _{n,(1-t)\varphi _ n+tV_{\theta _ n}}. \end{split} \]
More explicitly, for fixed $k$ the left-hand side is the value at $t$ of a polynomial $Q_ k(t)$ of degree at most $n$, and the right-hand side is the value of a polynomial $Q(t)$ of degree at most $n$. Choose $n+1$ distinct numbers $t_0,\ldots ,t_ n\in (0,1)$. Lagrange interpolation expresses $Q_ k(0)-Q(0)$ as a fixed linear combination of the numbers $Q_ k(t_ a)-Q(t_ a)$, which tend to $0$. Hence the same formula holds at $t=0$. From this, (16.9.31) follows.
Step 2. We now assume that the assumptions mentioned at the beginning of Step 1 are met. Note that we may assume that $\varphi _ j^ k$, $\varphi _ j$ are model potentials for all $j=1,\ldots ,n$, $k>0$ by proposition 16.9.6 and corollary 16.3.20.
We first handle the following monotone situation. Assume that, for each fixed $i$, the sequence $(\varphi _ i^ k)_ k$ is either increasing or decreasing. Let $\tilde{\varphi }_ i$ denote its pointwise limit, namely
\[ \tilde{\varphi }_ i=\inf _{k>0}\varphi _ i^ k \]
in the decreasing case and
\[ \tilde{\varphi }_ i=\operatorname *{sup}_{k>0}{}\! {}^{*}\varphi _ i^ k \]
in the increasing case. We assume moreover that $\tilde{\varphi }_ i\sim _ P\varphi _ i$ for every $i$. Since mixed non-pluripolar masses are invariant under replacing each entry by a $P$-equivalent potential, it suffices in this monotone situation to prove convergence to the mixed mass of the $\tilde{\varphi }_ i$'s.
After reordering the indices, we may assume that there is $i_0\in \{ 0,\ldots ,n\} $ such that $(\varphi _ i^ k)_ k$ is decreasing for $i\leq i_0$ and increasing for $i>i_0$. Therefore, for each $k>0$, using theorem 13.13.11, we have
\[ \int _ X \theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k}\geq \int _ X \theta _{1,\tilde{\varphi }_1}\wedge \cdots \wedge \theta _{i_0,\tilde{\varphi }_{i_0}}\wedge \theta _{i_0+1,\varphi ^ k_{i_0+1}}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k}. \]
Using corollary 13.13.12, we therefore conclude that
\[ \varliminf _{k\to \infty }\int _ X \theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k}\geq \int _ X \theta _{1,\tilde{\varphi }_1}\wedge \cdots \wedge \theta _{n,\tilde{\varphi }_ n}. \]
It remains to prove
DDD39F87B553
16.9.32
\begin{equation} \varlimsup _{k\to \infty }\int _ X \theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k}\leq \int _ X \theta _{1,\tilde{\varphi }_1}\wedge \cdots \wedge \theta _{n,\tilde{\varphi }_ n}. \end{equation}
By theorem 13.13.11, for each $k>0$, we have
\[ \int _ X \theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k}\leq \int _ X \theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{i_0,\varphi _{i_0}^ k}\wedge \theta _{i_0+1,\tilde{\varphi }_{i_0+1}}\wedge \cdots \wedge \theta _{n,\tilde{\varphi }_{n}}. \]
When proving (16.9.32), we may replace $\varphi _ j^ k$ by $\tilde{\varphi }_ j$ whenever $j>i_0$, $k>0$. Thus, we are reduced to the case where for all $i$, $(\varphi _ i^ k)_ k$ is decreasing.
By proposition 16.4.4, the masses $\int _ X\theta _{i,\varphi _ i^ k}^ n$ decrease to $\int _ X\theta _{i,\tilde{\varphi }_ i}^ n$. Hence, thanks to 13.13.30, for each $i=1,\ldots ,n$, we may take an increasing sequence $(b^ k_ i)_ k$ tending to $\infty $ satisfying
\[ b_ i^ k\in \left(1, \left( \frac{\int _ X \theta _{i,\varphi _ i^ k}^ n}{\int _ X \theta _{i,\varphi _ i^ k}^ n-\int _ X \theta _{i,\tilde{\varphi }_ i}^ n} \right)^{1/n} \right) \]
where the upper endpoint is understood as $\infty $ if the denominator is $0$. For each such $b_ i^ k$, 13.13.30 gives $\rho _ i^ k\in \mathrm{PSH}(X,\theta _ i)$ with
\[ (b^ k_ i)^{-1}\rho _ i^ k+\left(1-(b^ k_ i)^{-1}\right)\varphi _ i^ k\leq \tilde{\varphi }_ i. \]
Then by theorem 13.13.11 again,
\[ \prod _{i=1}^ n\left(1-(b^ k_ i)^{-1}\right)\int _ X \theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k}\leq \int _ X \theta _{1,\tilde{\varphi }_1}\wedge \cdots \wedge \theta _{n,\tilde{\varphi }_ n}. \]
Letting $k\to \infty $, we conclude (16.9.32). This proves the monotone situation.
We now return to the general case. It suffices to prove that any subsequence of
\[ \int _ X \theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k} \]
has a further subsequence converging to the desired limit. Starting with an arbitrary subsequence, we pass to a further subsequence, still indexed by $k$, so that the construction in proposition 16.9.17 applies simultaneously to the finitely many sequences $(\varphi _ i^ k)_ k$, $i=1,\ldots ,n$.
Thus, after discarding finitely many terms and relabeling, for each $i$ there are model potentials $\eta _ i^ k,\psi _ i^ k\in \mathrm{PSH}(X,\theta _ i)$ such that $(\eta _ i^ k)_ k$ is increasing, $(\psi _ i^ k)_ k$ is decreasing, and
\[ \eta _ i^ k\leq \varphi _ i^ k\leq \psi _ i^ k,\quad \eta _ i^ k\xrightarrow {d_ S}\varphi _ i,\quad \psi _ i^ k\xrightarrow {d_ S}\varphi _ i. \]
For the upper sequence, we take $\psi _ i^ k=P_{\theta _ i}[\operatorname *{sup}_{\ell \geq k}{}\! {}^{*}\varphi _ i^\ell ]$; as in (16.9.21), this gives $\inf _ k\psi _ i^ k=\varphi _ i$. For the lower sequence, the construction in proposition 16.9.17 together with proposition 16.4.3 shows that the $\eta _ i^ k$'s are model after discarding finitely many terms, and the proof of proposition 16.9.17 shows that their increasing limit is $P$-equivalent to $\varphi _ i$.
By theorem 13.13.11, for every $k$,
\[ \int _ X\theta _{1,\eta _1^ k}\wedge \cdots \wedge \theta _{n,\eta _ n^ k} \leq \int _ X\theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k} \leq \int _ X\theta _{1,\psi _1^ k}\wedge \cdots \wedge \theta _{n,\psi _ n^ k}. \]
The monotone situation just proved applies to both the lower and upper sequences, and both limiting mixed masses are equal to
\[ \int _ X\theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}. \]
The squeeze theorem therefore gives the desired convergence along this further subsequence. This concludes the proof.
$\square$
Pome
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