Pome

Theorem 8.11.14945FEE7DBDC5

Theorem 8.11.14.945FEE7DBDC5 Suppose $a_ j\downarrow a$ and $u_ i^ j\downarrow u_ i$ are bounded canonical global admissible potentials with common bounds. Then, for $1\leq p\leq m$,

\[ a_ j Q_ p(u_1^ j,\ldots ,u_ p^ j) \longrightarrow a Q_ p(u_1,\ldots ,u_ p) \]

weakly as currents with measure coefficients. At $p=m$ this is weak convergence of weighted Radon measures.

Proof. Work on a compact coordinate test set. By 8.11.12, every coefficient measure of the positive shifted currents converges weakly and has a uniform local variation bound. 8.11.13 strengthens this to domination by $C c_ m^*$ on Borel sets. 8.11.10 makes the canonical weights $a_ j$ and $a$ quasicontinuous for this capacity. Apply 8.11.11 with $s=m$ to each coefficient, and then use a finite partition of unity. At $p=m$, $Q_ m$ is a positive Radon measure, so this also gives the stated weighted measure convergence. $\square$

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