PomeTheorem 18.36.3.87C196C13DDE Under these hypotheses,
\[ \frac{q!}{m^ q}B_{X,m}\, \mu _ X \rightharpoonup \mathcal L\bigl (\mathrm{MA}_\theta (E)\bigr ). \]
If $n>q$, then $(n!/m^ n)B_{X,m}\, \mu _ X\to 0$ in total variation.
Theorem 18.36.387C196C13DDE
Proof. Apply [Theorem 14.3.1, SGPT] on the compact Kähler manifold $B$, with trivial auxiliary twisting line bundle. Its hypotheses are exactly our assumptions: $\psi $ has positive mass, $K$ is closed and non-pluripolar, $w$ is continuous, and $\mu $ is a Bernstein–Markov probability measure. It gives
\[ \frac{q!}{m^ q}B_{B,m}\, \mu \rightharpoonup \mathrm{MA}_\theta (E). \]
By proposition 18.36.1, the corresponding measures on $X$ are the lifts of these measures, and lifting preserves weak convergence; this proves the first assertion. In particular the $q$-normalized total masses are bounded, and multiplying them by $(n!/q!)m^{q-n}\to 0$ proves the last assertion when $n>q$. $\square$
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