Theorem 8.9.6.846671D5582A For $1\leq p\leq m$, $Q_ p$ is a positive current of order zero, and
\[ \int _ KQ_ p\wedge \omega ^{m-p} \leq C_{K,U,\omega ,p}(1+M)^ p. \]
It is the iterated limit, in the order of (8.9.4), of
\[ (\omega +\mathrm{dd}^{\mathrm{c}}u_1^{j_1})\wedge \cdots \wedge (\omega +\mathrm{dd}^{\mathrm{c}}u_ p^{j_ p})\wedge \omega ^{n-m}. \]
In particular $Q_ m$ is a positive Radon measure. These currents are local in the canonical potentials and independent of the chosen approximations for the fixed ordering. When all potentials are continuous, $Q_ p$ equals the continuous mixed product of 8.7.1.
Proof.
Expand the smooth shifted product over its subsets. Applying (8.9.4) to each subset, in its inherited ordering, gives (8.9.5). The absent indices have no effect on a subset's limit.
Each smooth product is positive: pairing with a decomposable strongly positive complementary test appends $m-p$ nonnegative rank-one $(1,1)$-forms, and polarized Gårding positivity applies. The smooth mixed mass bound is uniform in all indices. Distributional limits preserve positivity and the local trace bound at every step. Positive currents have order zero, so this gives the assertions for $Q_ p$, including the top-degree measure.
The recursion uses only bounded multiplication of canonical values and local distributional differentiation. It is therefore local. Its definition contains no approximation choices, so the proved limit is independent of those choices while the ordering is fixed. For continuous potentials, the approximations converge uniformly by Dini's theorem. The telescoping estimate in 8.7.1 is uniform in all remaining indices and shows that their iterated limit is the same continuous mixed product. This also proves permutation invariance in the continuous case.
$\square$
Pome
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