Proposition 8.12.19989CBEEE646

Proof. Choose smooth uniformly bounded decreasing approximations of all the potentials by 8.6.6. Construct $w_ j$ by 8.2.9. Work on one compact coordinate test set. Fix $\delta >0$ and a smooth $\chi _\delta :\mathbb R\to [0,1]$ that vanishes for $s\leq \delta /2$ and equals one for $s\geq \delta $. For $j$ large enough that $\epsilon _ j<\delta /2$, ordinary smooth locality gives

Indeed, wherever the weight is nonzero, $w_ j=u_ j$ on an open neighborhood. The equality elsewhere follows from its zero weight.

Both sequences of positive currents converge by 8.11.12; every coefficient variation is uniformly dominated by $C c_ m^*$ by 8.11.13. The potentials and their limits are quasicontinuous by 8.11.10. Apply 8.11.11 with $s=m$ and $F(x,y)=\chi _\delta (x-y)$ to the real and imaginary parts of each coefficient. Passing to the limit gives the same identity with $u,v,w,t_2,\ldots ,t_ p$. Choose the cutoffs so that $\chi _\delta (s)\to \mathbf1_{\{ s>0\} }$ pointwise as $\delta \downarrow 0$. Dominated convergence against the fixed finite coefficient variations on the test set gives the asserted restriction. A partition of unity proves it globally. $\square$