Proposition 13.2.67F7123F92478

Proof. In this proof we write $P^{(m)}(w_1,\ldots ,w_ p)=\mathrm{dd}^{\mathrm{c}}\max \{ w_1,-m\} \wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\max \{ w_ p,-m\} $ for plurisubharmonic $w_1,\ldots ,w_ p$ and an integer $m\geq 1$.

Symmetry and locality. Each truncated product $S_ k$ is symmetric in its factors, hence so is the limit. Suppose that $u_ j=v_ j$ on the plurifine open set $O$ for every $j$, and that both products are locally finite. By lemma 13.2.2, plurifine open subsets of $U$ are measurable for the completions of the total variations of the coefficient measures of both products, so the restrictions of the products to $O$ and to the plurifine open sets $W_ k=O\cap \bigcap _ j\{ u_ j>-k\} =O\cap \bigcap _ j\{ v_ j>-k\} $ are well defined; the sets $W_ k$ increase with $k$. On $W_ k$ the truncations $\max \{ u_ j,-k\} $ and $\max \{ v_ j,-k\} $ coincide, so the two level-$k$ products agree on $W_ k$ by theorem 5.47.22. Since $W_ k$ is contained in the level-$k$ sets of both tuples, lemma 13.2.2 shows that the two non-pluripolar products also agree on $W_ k$. The union of the $W_ k$ is $O$ minus the union of the pole sets of the $u_ j$, a pluripolar set, which has outer measure zero for these total variations by the same lemma. Letting $k\to \infty $, we conclude that the two products have the same restriction to $O$.

Dependence on the currents $\mathrm{dd}^{\mathrm{c}}u_ j$. Suppose that $\mathrm{dd}^{\mathrm{c}}u_ j=\mathrm{dd}^{\mathrm{c}}v_ j$ for every $j$. The distribution $v_ j-u_ j$ then has zero Laplacian, so by Weyl's lemma (theorem 3.1.11) it is a smooth function $w_ j$, and its complex Hessian vanishes: $w_ j$ is pluriharmonic. The plurisubharmonic functions $v_ j$ and $u_ j+w_ j$ agree almost everywhere, hence everywhere, since both are subharmonic (theorem 5.9.1, theorem 3.4.12). Fix a relatively compact open set $V\subset U$ and an integer $C$ such that $|w_ j|\leq C$ on $V$ for all $j$. Write $O_ k(u),S_ k(u)$ and $O_ k(v),S_ k(v)$ for the level sets and truncated products of the two tuples. On $V\cap O_ k(v)$ we have $u_ j=v_ j-w_ j>-k-C$, hence

The functions $\max \{ v_ j,-k\} $ and $\max \{ u_ j,-k-C\} +w_ j$ are bounded and plurisubharmonic on $V$ and coincide on $V\cap O_ k(v)$. Moreover, adding the pluriharmonic function $w_ j$ to a factor does not change Bedford–Taylor products. Indeed, by additivity (proposition 7.2.2) and symmetry (theorem 7.3.7), the difference is a sum of products having $\mathrm{dd}^{\mathrm{c}}w_ j$ as first factor; by proposition 7.2.2(1), such a product is the exterior product of the remaining closed positive current with the smooth form $\mathrm{dd}^{\mathrm{c}}w_ j=0$. As $V\cap O_ k(v)\subset O_{k+C}(u)$, plurifine locality gives on $V$

Thus (13.2.3) for the $u_ j$ implies (13.2.3) for the $v_ j$ on compact subsets of $V$, and exchanging the two tuples (that is, replacing $w_ j$ by $-w_ j$) gives the converse. If both products are locally finite, the display and lemma 13.2.2 give $\mathbf1_{O_ k(v)}\langle \mathrm{dd}^{\mathrm{c}}v_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ p\rangle =\mathbf1_{O_ k(v)}\langle \mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ p\rangle $ on $V$ for every $k$. The sets $O_ k(v)$ exhaust the complement of the union of the pole sets, which are the same for the two tuples and carry no mass for either product. Hence the two products coincide.

Additivity. Let $u_1'$ be plurisubharmonic and put

These plurifine open sets increase with $k$ and exhaust the complement of the union of the pole sets of $u_1,u_1',u_2,\ldots ,u_ p$. On $E_ k$ we have $\max \{ u_1+u_1',-2k\} =u_1+u_1'=\max \{ u_1,-k\} +\max \{ u_1',-k\} $ and $\max \{ u_ j,-2k\} =\max \{ u_ j,-k\} $ for $j\geq 2$. Plurifine locality and the additivity of bounded products (proposition 7.2.2) therefore give

Before local finiteness is known we argue with trace measures. By proposition 5.47.23, for each of the three tuples the trace measures of the truncated products increase to a Borel measure, possibly infinite on compact sets, for which every pluripolar set has outer measure zero; for a locally finite product it is the trace measure of the product. The set $E_ k$ is contained in the level-$2k$ set of the first tuple and in the level-$k$ sets of the other two, so by the identity $\mathbf1_{O_ k}S_ l=S_ k$ of lemma 13.2.2, the restriction of each limit measure to $E_ k$ is the restriction of the corresponding trace measure in (13.2.7). Hence, on $E_ k$, the limit measure of the first tuple is the sum of those of the other two. Letting $k\to \infty $, monotone convergence gives this equality on every Borel subset of $\bigcup _ kE_ k$, and hence everywhere, because the complement of $\bigcup _ kE_ k$ is pluripolar. Since all three measures are positive, the first is locally finite if and only if the other two are; this is the asserted equivalence of local finiteness.

When the products are locally finite, (13.2.7) and lemma 13.2.2 give the additivity identity on each $E_ k$, and letting $k\to \infty $ proves it. For $c>0$ the same argument applies to the sets $\{ u_1>-k\} \cap \{ cu_1>-k\} \cap \bigcap _{j\geq 2}\{ u_ j>-k\} $, on which $\max \{ cu_1,-k\} =c\max \{ u_1,-k\} $, and proves the homogeneity assertion. $\square$