Theorem 8.11.12C044CD1C65C4

Proof. Work on a compact coordinate test set. Degree zero is constant. Suppose the result holds for $D_{p-1}$, which has measure coefficients since $p-1<m$. The preceding two lemmas, applied coefficientwise with weight $u_ p^ j$, give convergence of $u_ p^ jD_{p-1}(u_1^ j,\ldots ,u_{p-1}^ j)$. Applying $\mathrm{dd}^{\mathrm{c}}$ proves distributional convergence at degree $p$. For $p<m$, the uniform coefficient bounds in 8.9.3 upgrade this to weak convergence of measures, by smooth approximation of compactly supported continuous tests. The same weighted lemma gives the additional weighted conclusion.

For each fixed list choose simultaneous smooth decreasing global approximations. Their smooth raw Hessian wedges are symmetric. The convergence just proved identifies their limit with $D_ p$ in any ordering, proving symmetry without an exchange of previously unproved iterated limits.

The subset expansion for $R_ p$ uses fixed smooth powers of $\omega $, so its distributional convergence and symmetry follow from those of $D_ I$. Its wedge $Q_ p$ is positive with uniform local masses by 8.9.6. Distributional convergence and these mass bounds imply weak convergence as an order-zero positive current, including the Radon measure conclusion at $p=m$. A partition of unity on a finite coordinate cover proves the global assertions. $\square$