13.4 Bounded factors and the first-degree productF483C973DB98

Proof. We treat $r\geq 2$ and smooth $V$ by an integration by parts on small balls, then general $V$ by monotone approximation, and reduce $r=1$ to $r=2$ by adding a variable.

Step 1: $r\geq 2$ and $V$ smooth. Here $\| v\| _{L^\infty (L)}=\sup _ L|v|$, as in theorem 7.2.5. If it vanishes, then $v=0$ on $L$, so $\mathrm{dd}^{\mathrm{c}}v=0$ on $\operatorname{Int}L\supset K$, and the left side vanishes. Otherwise both sides are homogeneous of degree one in $v$, and the left side does not change when a constant is added to $v$; replacing $v$ by $v/(2\| v\| _{L^\infty (L)})-3/2$, it therefore suffices to prove that $\int _ K(-V)\, \mathrm{dd}^{\mathrm{c}}v\wedge T\wedge \beta ^{r-1}\leq C\int _ L(-V)T\wedge \beta ^ r$ for plurisubharmonic functions $v$ satisfying $-2\leq v\leq -1$ on $\operatorname{Int}L$. Cover $K$ by finitely many balls $B(a,R_1)$ such that $\overline{B(a,R)}\subset \operatorname{Int}L$ for some $R>R_1$; it suffices to bound the integral over one ball $B(a,R_1)$.

Put $B=B(a,R)$, $\psi =|z-a|^2-R^2$ and $b=\mathrm{dd}^{\mathrm{c}}\psi =\beta /\pi $. Fix $R_1<R_2<R$, and let $\widetilde v=\max \{ v,A\psi \} $, with $A=2/(R^2-R_2^2)$, be the collar modification (7.3.2) of $v$ on $\operatorname{Int}L$, with $N=2$ and with $R^2-R_2^2$ in place of $\delta $. Thus $\widetilde v$ is plurisubharmonic on $\operatorname{Int}L$ and takes values in $[-2,0)$ on $B$; it equals $v$ on $B(a,R_2)$, and $A\psi $ on the collar $\{ R^2-1/A<|z-a|^2<R^2\} $. Choose $0<\delta <1/A$ and put $B_\delta =\{ \psi <-\delta \} $, so that $\widetilde v=A\psi $ on $B\setminus B_\delta $ and $B(a,R_1)\subset B_\delta $. Choose a smooth function $\kappa $ with $0\leq \kappa \leq R^2$, with compact support in $B$ and equal to $-\psi $ on a neighborhood of $\overline{B_\delta }$; this is possible because $0<-\psi \leq R^2$ on $B$. Put $S=\mathrm{dd}^{\mathrm{c}}\widetilde v\wedge T$ and $Q=\mathrm{dd}^{\mathrm{c}}V\wedge T$; since $V$ is smooth and plurisubharmonic, $Q$ is a closed positive current of bidimension $(r-1,r-1)$. Since $\mathrm{dd}^{\mathrm{c}}\kappa =-b$ near $\overline{B_\delta }$ and $S=Ab\wedge T$ on $B\setminus B_\delta $,

For the second line we integrate by parts against the smooth compactly supported function $\kappa $: since $V$ is smooth and $\mathrm{dd}^{\mathrm{c}}\widetilde v\wedge T\wedge b^{r-2}$ is closed, $\mathrm{dd}^{\mathrm{c}}(V\mathrm{dd}^{\mathrm{c}}\widetilde v\wedge T\wedge b^{r-2})=\mathrm{dd}^{\mathrm{c}}V\wedge \mathrm{dd}^{\mathrm{c}}\widetilde v\wedge T\wedge b^{r-2}$, and the Bedford–Taylor product is symmetric with respect to the smooth factor.

We bound the two terms on the right. Choose a compact set $L'$ with $\overline B\subset \operatorname{Int}L'$ and $L'\subset \operatorname{Int}L$. Since $0\leq \kappa \leq R^2$, the Chern–Levine–Nirenberg inequality (theorem 7.2.5) applied to $Q$ bounds the first term by a constant times the trace mass of $Q$ on $\overline B$. To bound this mass, test $Q$ against a smooth cutoff $0\leq \chi \leq 1$ equal to one on $\overline B$ and supported in $L'$:

since $V\leq 0$ and the coefficients of $T\wedge \mathrm{dd}^{\mathrm{c}}\chi \wedge \beta ^{r-1}$ are bounded by a fixed multiple of the trace measure. The collar term has the same bound, because $\mathrm{dd}^{\mathrm{c}}\kappa $ is a fixed smooth form. Finally, $S=\mathrm{dd}^{\mathrm{c}}v\wedge T$ on $B(a,R_1)$ and $-V\geq 0$, so the left side of the display bounds $\pi ^{1-r}\int _{B(a,R_1)}(-V)\mathrm{dd}^{\mathrm{c}}v\wedge T\wedge \beta ^{r-1}$. Summing over the finitely many balls proves the estimate, with a constant depending only on the chosen radii.

Step 2: $r\geq 2$ and $V$ arbitrary. Choose a compact set $L_1$ with $K\subset \operatorname{Int}L_1$ and $L_1\subset \operatorname{Int}L$, depending only on $K$ and $L$. The mollifications $V_ l=V*\rho _{\varepsilon _ l}$, with $\varepsilon _ l\downarrow 0$, are defined, smooth and plurisubharmonic on a neighborhood of $L_1$ for large $l$, and they decrease to $V$ (proposition 5.9.2(2)); in particular $V\leq V_ l\leq 0$, so that $0\leq -V_ l\leq -V$. Step 1 only involves the weight on a neighborhood of the larger compact set, so it applies to $V_ l$ and the pair $(K,L_1)$ and gives

As $l\to \infty $, $-V_ l$ increases to $-V$, and monotone convergence against the positive measure $\mathrm{dd}^{\mathrm{c}}v\wedge T\wedge \beta ^{r-1}$ proves the estimate.

Step 3: $r=1$. Let $t$ be the coordinate of $\mathbb C$, put $\beta _ t=\frac{\mathrm i}{2}\mathrm dt\wedge \mathrm d\bar t$, let $D'\Subset D''\Subset D$ be the discs about $0$ of radii $1,2,3$, and let $P\colon U\times D\to U$ be the projection. By proposition 5.6.2, $P^*T$ is a closed positive current of bidimension $(2,2)$ on $U\times D$. Moreover, by corollary 5.6.4(1), the pullback of a current $R=\sum R_{IJ}\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$ of order zero on $U$ is $P^*R=\sum (R_{IJ}\otimes \lambda _ D)\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$, where $\lambda _ D$ is Lebesgue measure on $D$. Since $v$ is a locally bounded Borel function, $P^*(vT)=(v\circ P)P^*T$ by corollary 5.6.4(2), and since $P^*$ commutes with $\partial $ and $\bar\partial $,

The functions $V\circ P\leq 0$ and $v\circ P$ are plurisubharmonic (proposition 5.9.3), and $\| v\circ P\| _{L^\infty (L\times \overline{D''})}=\| v\| _{L^\infty (L)}$. We apply Step 2 with $r=2$ to these data, to the Euclidean form $\beta +\beta _ t$ and to the compact sets $K\times \overline{D'}\subset \operatorname{Int}L\times D''=\operatorname{Int}(L\times \overline{D''})$. Since $T\wedge \beta ^2=0$ and $\mathrm{dd}^{\mathrm{c}}v\wedge T\wedge \beta =0$ for bidegree reasons, the trace measures in Step 2 are

that is, the products of the measures $2T\wedge \beta $ and $\mathrm{dd}^{\mathrm{c}}v\wedge T$ with Lebesgue measure on $D$. By Fubini's theorem (proposition 2.2.7), the inequality of Step 2 therefore reads

where $C$ is the constant of Step 2 for $K\times \overline{D'}$ and $L\times \overline{D''}$ in dimension $n+1$, which depends only on $K$, $L$ and $n$. This is the case $r=1$, with $C_{K,L}=8C$. $\square$

Proof. Let $E$ be a pluripolar subset of the coordinate domain $U$ on which $T$ lives. By definition 5.47.2 and since $U$ has a countable basis, $E$ is covered by countably many compact sets $K$, each having a compact neighborhood $L$ with $K\subset \operatorname{Int}L$, together with an open neighborhood $W$ of $L$ and a function $V\in \mathrm{PSH}(W)$ such that $E\cap K\subset \{ V=-\infty \} $; shrinking $W$ and subtracting a constant, we may assume $V\leq 0$. The set $\{ V=-\infty \} $ is a pluripolar Borel set, so it suffices to show that $K\cap \{ V=-\infty \} $ has measure zero for the trace measure of $\mathrm{dd}^{\mathrm{c}}v\wedge T$.

Put $\mu =\mathbf1_ LT\wedge \beta ^ r$. This is a finite measure that does not charge $\{ V=-\infty \} $, so $V$ is finite $\mu $-almost everywhere. Hence $\min \{ -V/R,1\} \to 0$ $\mu $-almost everywhere as $R\to \infty $, and by dominated convergence we may choose numbers $R_ j\geq 2^ j$ such that

Put $\widetilde V=\sum _{j\geq 1}\max \{ V/R_ j,-1\} $. The partial sums are plurisubharmonic by proposition 5.9.5, and they decrease, because every term is nonpositive. Their limit satisfies $\widetilde V\geq \sum _ jV/R_ j\geq V$, since $V\leq 0$ and $\sum _ j1/R_ j\leq 1$; hence it is not identically $-\infty $ on any component of $W$, and it is plurisubharmonic by theorem 5.10.1. Moreover $\widetilde V\leq 0$, $\widetilde V=-\infty $ on $\{ V=-\infty \} $, where every term equals $-1$, and by monotone convergence

By lemma 13.4.1, applied on $W$, the function $-\widetilde V$ is integrable on $K$ for the trace measure $\mathrm{dd}^{\mathrm{c}}v\wedge T\wedge \beta ^{r-1}$ of $\mathrm{dd}^{\mathrm{c}}v\wedge T$. Hence this measure gives zero mass to $K\cap \{ \widetilde V=-\infty \} \supset K\cap \{ V=-\infty \} $, which proves the first assertion. The second follows by induction on $q$, since $\mathrm{dd}^{\mathrm{c}}v_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ q\wedge T=\mathrm{dd}^{\mathrm{c}}v_1\wedge (\mathrm{dd}^{\mathrm{c}}v_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ q\wedge T)$, where the base current in parentheses is closed and positive of bidimension $(r-q+1,r-q+1)$. $\square$

Proof. For $q=0$ there is nothing to prove. Inserting the bounded factors $u_ q,u_{q-1},\ldots ,u_1$ one at a time, we reduce to the case $q=1$. So let $v=u_1$ be locally bounded and $S=\langle \mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ p\rangle $ locally finite. By theorem 13.3.1, $S$ is a closed positive current, so $R=\mathrm{dd}^{\mathrm{c}}v\wedge S$ is defined; it is closed and positive, and by lemma 13.4.2 it gives zero mass to pluripolar sets, because $S$ does. If $p=1$, then $S=1$ and the assertion is that of bounded Bedford–Taylor theory. Let $p\geq 2$. The assertion is local, so we may assume $u_ j\leq 0$ for $j\geq 2$ and put

The functions $w_ k$ are plurisubharmonic by proposition 5.9.5, and they take values in $[0,1]$ because all $u_ j\leq 0$. They vanish outside $O_ k$: if $u_ i\leq -k$ for some $i\geq 2$, then $\sum _{j\geq 2}u_ j\leq u_ i\leq -k$. The sets $\{ w_ k>0\} =\{ \sum _{j\geq 2}u_ j>-k\} $ increase to the set where all $u_ j$, $j\geq 2$, are finite.

Take smooth plurisubharmonic functions $v_ l$ decreasing to $v$ near a given compact set. By lemma 13.2.2, $S$ and $\Theta _ k$ have the same restriction to $O_ k$; since $w_ k$ vanishes outside $O_ k$ and $\mathrm{dd}^{\mathrm{c}}v_ l$ is smooth,

The weighted form of theorem 7.3.7, applied with the constant sequence of weights $w_ k$ and the fixed base currents $S$ on the left and $\Theta _ k$ on the right, gives $w_ kR=w_ k\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ k$ in the limit $l\to \infty $. Multiplying by the bounded Borel functions $\mathbf1_{\{ w_ k>\varepsilon \} }/w_ k$ and letting $\varepsilon \downarrow 0$, we see that $R$ and $\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ k$ have the same restriction to $\{ w_ k>0\} $. Now fix an integer $m\geq 1$. For $k\geq m$, the bounded functions $\max \{ u_ i,-k\} $ and $\max \{ u_ i,-m\} $ coincide on $O_ m$, so $\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ k=\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ m$ on $O_ m$ by theorem 5.47.22. Since the sets $\{ w_ k>0\} \cap O_ m$, $k\geq m$, exhaust $O_ m$, we conclude that $\mathbf1_{O_ m}R=\mathbf1_{O_ m}\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ m$.

Consider finally the full tuple $v,u_2,\ldots ,u_ p$ on a relatively compact open set on which $|v|<m_0$. For $m\geq m_0$, its level-$m$ set is $O_ m$ and $v$ needs no truncation, so its truncated product at level $m$ is $\mathbf1_{O_ m}\mathrm{dd}^{\mathrm{c}}v\wedge \Theta _ m=\mathbf1_{O_ m}R$. These restrictions are bounded by $R$, hence have locally bounded mass, and they increase to $R$ because $R$ gives the pole sets of $u_2,\ldots ,u_ p$ zero mass. Therefore the non-pluripolar product of $v,u_2,\ldots ,u_ p$ is locally finite and equals $R=\mathrm{dd}^{\mathrm{c}}v\wedge S$. $\square$

Proof. Fix open sets $Q''\Subset Q$ and $D''\Subset D$, so that $\mu _ z(Q'')<\infty $ for every $z$. If $A\subset Q''$ is open, the continuous functions $a_ l(x)=\min \{ 1,\max \{ l\operatorname {dist}(x,\mathbb C^ r\setminus A)-1,0\} \} $ have compact support in $A$ and increase to $\mathbf1_ A$, so $z\mapsto \mu _ z(A)$ is Borel by monotone convergence. Let $\mathcal D$ be the family of Borel sets $A\subset Q''$ for which $z\mapsto \mu _ z(A)$ is Borel on $G$. It contains the open subsets of $Q''$, which are closed under finite intersections and generate the Borel $\sigma $-algebra of $Q''$. It is closed under complements in $Q''$, because $\mu _ z(Q''\setminus A)=\mu _ z(Q'')-\mu _ z(A)$ with finite values, and under countable disjoint unions. By Dynkin's theorem (lemma 2.2.6), $\mathcal D$ contains every Borel subset of $Q''$.

Next, let $\mathcal D'$ be the family of Borel sets $E\subset Q''\times D''$ for which $z\mapsto \mu _ z(E_ z)$ is Borel on $G$. It contains the products $A\times A'$ of Borel sets $A\subset Q''$ and $A'\subset D''$, since $\mu _ z((A\times A')_ z)=\mu _ z(A)\mathbf1_{A'}(z)$; these products are closed under finite intersections and generate the Borel $\sigma $-algebra of $Q''\times D''$, because every open subset of $Q''\times D''$ is a countable union of products of open balls. As before, $\mathcal D'$ is closed under complements and countable disjoint unions, so it contains every Borel subset of $Q''\times D''$. Exhausting $Q$ and $D$ by increasing sequences of such sets $Q''$ and $D''$, and using monotone convergence, we obtain the first assertion.

By the first assertion, $\lambda (E)$ is defined, and $\lambda $ is countably additive by monotone convergence, applied first to the measures $\mu _ z$ and then to the integral over $G$. The formula for $\int h\, \mathrm d\lambda $ holds for indicator functions of Borel sets by definition, hence for nonnegative simple functions by linearity, and for nonnegative Borel functions by monotone convergence, applied in the same order; the inner integrals are Borel in $z$ as increasing limits of Borel functions. $\square$

Proof. If $r=0$ or $m=0$, one factor of $Q\times D$ is a point, carrying counting measure, and the corresponding product is the empty product $1$; then the two sides of (13.4.6) are the same integral. Let $r,m\geq 1$. For smooth potentials the formula is Fubini's theorem; we pass to the limit for continuous weights $h$, and then extend the formula to Borel weights by the uniqueness of measures.

Approximation. Let $Q'\Subset Q$ and $D'\Subset D$ be open. For $\varepsilon _ l\downarrow 0$, the mollifications $u_{j,l}=u_ j*\rho _{\varepsilon _ l}$ are smooth and plurisubharmonic on a neighborhood of $\overline{Q'\times D'}$ for large $l$, they decrease to $u_ j$ (proposition 5.9.2(2)), and they are bounded there by a constant $B$, since $u_ j\leq u_{j,l}\leq u_{j,l_0}$. Let $M^{(l)}$ and $M_ z^{(l)}$ be the corresponding products. By theorem 7.3.7, $M^{(l)}\to M$ weakly on $Q'\times D'$, and for every $z\in D'$, $M_ z^{(l)}\to M_ z$ weakly on $Q'$, since $u_{j,l}(\, \cdot \, ,z)$ decreases to $u_ j(\, \cdot \, ,z)$. By theorem 7.2.5, for every compact set $A\subset Q'$ the masses $M_ z^{(l)}(A)$ and $M_ z(A)$ are bounded by a constant depending only on $A$, $Q'$ and $B$, uniformly in $l$ and $z\in D'$; similarly for $M^{(l)}$ and $M$ on compact subsets of $Q'\times D'$. For positive measures with locally uniformly bounded masses, weak convergence implies convergence of the integrals of continuous functions with compact support, because such functions are uniform limits of test functions supported in a fixed compact set (lemma 2.5.1).

Continuous weights. Let $\eta =f\, \mathrm dV_ z\geq 0$, where $\mathrm dV_ z$ is the Euclidean volume form of $\mathbb C^ m$ and $f\geq 0$ is smooth, and let $h$ be continuous with compact support in $Q'\times D'$. In the smooth exterior product $M^{(l)}$, wedging with the form $p_ D^*\eta $ of top degree in $z$ kills every term containing a differential $\mathrm dz_ i$ or $\mathrm d\bar z_ i$, so that $p_ D^*\eta \wedge M^{(l)}=f(z)\, M_ z^{(l)}(x)\wedge \mathrm dV_ z$, and Fubini's theorem (proposition 2.2.7) gives (13.4.6) for the smooth approximations. As $l\to \infty $, the left side converges to $\int h\, p_ D^*\eta \wedge M$, by the approximation step applied to the positive measures $p_ D^*\eta \wedge M^{(l)}$. On the right, the inner integrals are continuous functions of $z$, they converge to $\int _ Qh(x,z)\, M_ z(x)$ for every $z$, and they are bounded by $\sup |h|$ times a constant, uniformly in $l$ and $z$; hence the limit is a Borel function of $z$, and dominated convergence proves (13.4.6). A finite smooth partition of unity (lemma 2.5.2), subordinate to a cover of $\operatorname{Supp}h$ by such products $Q'\times D'$, treats arbitrary continuous $h$ with compact support.

Measurability. For a continuous function $a$ with compact support in $Q$, the function $z\mapsto \int a\, M_ z$ is Borel on $D$: if the support of $a$ lies in $Q'$, then on $D'$ it is the pointwise limit of the continuous functions $z\mapsto \int a\, M_ z^{(l)}$, and $D$ is covered by countably many such sets $D'$. Hence lemma 13.4.4, applied with $G=D$ and $\mu _ z=M_ z$, shows that $z\mapsto \int _ Qh(x,z)\, M_ z(x)$ is Borel for every nonnegative Borel function $h$, and hence for every bounded Borel function $h$ with compact support.

Borel weights. Let $\eta =f\, \mathrm dV_ z\geq 0$ as before, and let $\lambda $ be the measure of lemma 13.4.4 for $\mu _ z=M_ z$ and this function $f$, so that

for Borel sets $E\subset Q\times D$. The uniform local bound for the fiber masses makes $\lambda $ finite on compact sets, so it is a Radon measure by proposition 2.2.2. By lemma 13.4.4, $\int h\, \mathrm d\lambda =\int _ D(\int _ Qh(x,z)\, M_ z(x))\, \eta (z)$ for every nonnegative Borel function $h$, hence for every continuous $h$ with compact support. By the step on continuous weights, $\lambda $ and $p_ D^*\eta \wedge M$ therefore have the same integral against every test function, so they are equal by the uniqueness in theorem 2.5.6. This proves the formula for nonnegative Borel $h$, and then for bounded real Borel $h$ with compact support, by splitting $h$ into its positive and negative parts.

Finally, let $\eta =f\, \mathrm dV_ z$ be a smooth real form and $h$ a bounded Borel function supported in $K\times K_ D$, with $K\subset Q$ and $K_ D\subset D$ compact. Choose $\kappa \in C_ c^\infty (D)$ with $0\leq \kappa \leq 1$ and $\kappa =1$ on $K_ D$, and put $c=\sup _{\operatorname{Supp}\kappa }|f|$. Replacing $\eta $ by $\kappa \eta $ changes neither side of (13.4.6), and $\kappa \eta =\kappa (f+c)\, \mathrm dV_ z-c\kappa \, \mathrm dV_ z$ is the difference of two smooth nonnegative forms. This proves the formula for signed $\eta $. $\square$

Proof. Upper bound. The assertion is local, so we may assume $u\leq 0$. The truncations $u_ l=\max \{ u,-l\} $ decrease to $u$ and converge to it in $L^1_{\mathrm{loc}}$ (theorem 5.10.1), so $\mathrm{dd}^{\mathrm{c}}u_ l\to \mathrm{dd}^{\mathrm{c}}u$ weakly. For fixed $k$ and every $l\geq k$, plurifine locality (theorem 5.47.22) and lemma 13.2.1 give $\mathbf1_{\{ u>-k\} }\mathrm{dd}^{\mathrm{c}}u_ k=\mathbf1_{\{ u>-k\} }\mathrm{dd}^{\mathrm{c}}u_ l\leq \mathrm{dd}^{\mathrm{c}}u_ l$, and letting $l\to \infty $ yields $\mathbf1_{\{ u>-k\} }\mathrm{dd}^{\mathrm{c}}u_ k\leq \mathrm{dd}^{\mathrm{c}}u$. Hence the truncated products have locally bounded mass, the non-pluripolar product $S=\langle \mathrm{dd}^{\mathrm{c}}u\rangle $ is locally finite, and $0\leq S\leq \mathrm{dd}^{\mathrm{c}}u$. Since $S$ gives no mass to the pole set of $u$, it follows that $S\leq \mathbf1_{\{ u>-\infty \} }\mathrm{dd}^{\mathrm{c}}u$. It remains to prove the reverse inequality.

One variable. Let $n=1$ and put $\mu =\mathrm{dd}^{\mathrm{c}}u$, a positive measure. Let $V$ be a relatively compact open subset of the domain, and choose a smooth compactly supported function $\kappa \geq 0$ equal to one on $V$. With $\nu =\kappa \mu $ and the fundamental solution $E_2$ of (3.1.14), theorem 3.4.14 gives

where $h$ is harmonic; the factor $4\pi $ appears because $\mathrm{dd}^{\mathrm{c}}u=(\Delta u)/(4\pi )$ as planar measures, while the Riesz measure is $\Delta u$. Put $p=E_2*\nu $ and $\nu _ f=\mathbf1_{\{ p>-\infty \} }\nu $; on $V$ we have $\{ p>-\infty \} =\{ u>-\infty \} $. The kernel $E_2$ is bounded above on the compact set of differences of points of $\operatorname{Supp}\nu $, so removing positive mass from $\nu $ lowers its potential on $\operatorname{Supp}\nu $ by at most a constant: $E_2*\nu _ f\geq p-C$ there. Hence $E_2*\nu _ f$ is finite $\nu _ f$-almost everywhere, and theorem 4.1.10 writes $\nu _ f=\sum _ i\nu _ i$ with compactly supported measures $\nu _ i$ whose potentials are finite and continuous.

For each $N$, the measure $\nu ^{(N)}=\sum _{i\leq N}\nu _ i$ has the continuous subharmonic potential $v_ N=4\pi E_2*\nu ^{(N)}$, with $\mathrm{dd}^{\mathrm{c}}v_ N=\nu ^{(N)}$. On $V$ we have $\mu =\nu \geq \nu _ f\geq \nu ^{(N)}$, so the distribution $u-v_ N$ has positive Laplacian on $V$. By theorem 3.4.12(3), this distribution has a subharmonic representative $s$ on $V$, and by part (2) of that theorem the value of $s$ at each point is the limit of its averages over small discs. These averages are also those of the function $u-v_ N$, and they converge to $u-v_ N$ at every point, by part (2) applied to $u$ and the continuity of $v_ N$. Hence $s=u-v_ N$ everywhere on $V$, and $u-v_ N$ is subharmonic on $V$. Since $v_ N$ is locally bounded, multilinearity (proposition 13.2.6) gives

Letting $N\to \infty $ yields $S\geq \nu _ f=\mathbf1_{\{ u>-\infty \} }\mu $ on $V$, which proves the proposition for $n=1$.

Several variables. Let $n\geq 2$. The assertion is local, so we may work on a polydisc $P$ such that $u\leq 0$ is plurisubharmonic on a neighborhood of $\overline P$. Fix one of the coordinates, denote it by $x$, and write $z\in \mathbb C^{n-1}$ for the others, so that $P=\Delta \times Z$ with a disc $\Delta \subset \mathbb C$ and a polydisc $Z\subset \mathbb C^{n-1}$. Let $\mathrm dV_ z$ be the Euclidean volume form of the $z$-coordinates, viewed as an $(n-1,n-1)$-form, and $\mathrm dA$ the area measure on $\Delta $. We disintegrate $\mathrm{dd}^{\mathrm{c}}u\wedge \mathrm dV_ z$ along the fibers $\Delta \times \{ z\} $ and apply the one-variable case on each fiber. Since $u$ is integrable on $\Delta \times Z$, Fubini's theorem (proposition 2.2.7) shows that $G=\{ z\in Z:\int _\Delta |u(x,z)|\, \mathrm dA(x)<\infty \} $ is a Borel set of full measure in $Z$. For $z\in G$ the function $u_ z=u(\, \cdot \, ,z)$ is integrable, hence not identically $-\infty $, so it is subharmonic on $\Delta $ by proposition 5.9.3; let $\mu _ z=\mathrm{dd}^{\mathrm{c}}_ xu_ z$ be its Hessian measure.

For a test function $a$ on $\Delta $, we have $\int a\, \mathrm d\mu _ z=\int _\Delta u_ z\mathrm{dd}^{\mathrm{c}}_ xa$, and Fubini's theorem, applied to the positive and negative parts of the integrand, shows that this is a Borel function of $z\in G$. Uniform approximation by test functions supported in a fixed compact set extends this to continuous functions $a$ with compact support, since each $\mu _ z$ is finite on compact sets. Hence lemma 13.4.4, with $f=1$, gives a measure $\lambda $ on $\Delta \times Z$ with $\int h\, \mathrm d\lambda =\int _ G(\int h(\, \cdot \, ,z)\, \mathrm d\mu _ z)\, \mathrm dV(z)$ for nonnegative Borel $h$. If $\chi \in C_ c^\infty (\Delta )$ satisfies $0\leq \chi \leq 1$ and equals one on a disc $\Delta '\Subset \Delta $, then $\mu _ z(\Delta ')\leq \int u_ z\mathrm{dd}^{\mathrm{c}}_ x\chi \leq C_\chi \int _\Delta |u_ z|\, \mathrm dA$, which is integrable over $G$; so $\lambda $ is finite on compact sets. For a test function $\varphi $ on $\Delta \times Z$, only the derivative $\partial _ x\partial _{\bar x}\varphi $ survives in $\mathrm{dd}^{\mathrm{c}}\varphi \wedge \mathrm dV_ z$, and Fubini's theorem gives

By the uniqueness in theorem 2.5.6, the positive measure $\mathrm{dd}^{\mathrm{c}}u\wedge \mathrm dV_ z$ equals $\lambda $, that is,

for every nonnegative Borel function $g$ on $\Delta \times Z$.

Now let $g\geq 0$ be a bounded Borel function with compact support in $\Delta \times Z$, and apply lemma 13.4.5, with $r=1$, $Q=\Delta $, $D=Z$ and $\eta =\mathrm dV_ z$, to the bounded function $u_ k=\max \{ u,-k\} $ and the weight $g\mathbf1_{\{ u>-k\} }$:

As $k\to \infty $, the left side increases to $\int gS\wedge \mathrm dV_ z$ by lemma 13.2.2, since $\mathbf1_{\{ u>-k\} }\mathrm{dd}^{\mathrm{c}}u_ k$ is the truncated product $S_ k$ of $u$. For $z\in G$, the inner integral on the right is the integral of $g(\, \cdot \, ,z)$ against the truncated product of $u_ z$ at level $k$; by lemma 13.2.2 and the one-variable case, it increases to $\int g\, \mathrm d\langle \mathrm{dd}^{\mathrm{c}}_ xu_ z\rangle =\int g\mathbf1_{\{ u_ z>-\infty \} }\, \mathrm d\mu _ z$. Since $Z\setminus G$ is a null set, monotone convergence in $z$ and (13.4.9), applied to $g\mathbf1_{\{ u>-\infty \} }$, give $\int gS\wedge \mathrm dV_ z=\int g\mathbf1_{\{ u>-\infty \} }\mathrm{dd}^{\mathrm{c}}u\wedge \mathrm dV_ z$, so that $S\wedge \mathrm dV_ z=\mathbf1_{\{ u>-\infty \} }\mathrm{dd}^{\mathrm{c}}u\wedge \mathrm dV_ z$.

Since $\beta ^{n-1}=(n-1)!\sum _{j=1}^ n\mathrm dV_{(j)}$, where $\mathrm dV_{(j)}$ is the volume form of the coordinates other than the $j$th, summing over the $n$ coordinate directions shows that the trace measures of $S$ and of $\mathbf1_{\{ u>-\infty \} }\mathrm{dd}^{\mathrm{c}}u$ coincide. Thus $\mathbf1_{\{ u>-\infty \} }\mathrm{dd}^{\mathrm{c}}u-S$ is a positive current with zero trace measure, and it vanishes by theorem 5.3.7. This proves (13.4.8); closedness follows from theorem 13.3.1. $\square$

Proof. In the affine chart put $\rho (z)=\log (1+|z|^2)$, so that $\omega _{\mathrm{FS}}=\mathrm{dd}^{\mathrm{c}}\rho $. The integrand is nonpositive, since $|z-a|^2\leq (|z|+|a|)^2\leq (1+|z|^2)(1+|a|^2)$ by the Cauchy–Schwarz inequality. On the affine chart,

Since $\log |z-a|^2=4\pi E_2(z-a)$ with the fundamental solution $E_2$ of (3.1.14), the function $p_\mu =4\pi E_2*\mu $ is subharmonic and locally integrable with $\Delta p_\mu =4\pi \mu $, by proposition 3.4.11. Since $\mathrm{dd}^{\mathrm{c}}=\Delta /(4\pi )$ on functions of one complex variable, as planar measures, this gives $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q=\mathrm{dd}^{\mathrm{c}}p_\mu =\mu $ on the affine chart, and $q=p_\mu -c-\rho $ is quasi-plurisubharmonic there.

Near infinity use the coordinate $w=1/z$, in which the integrand becomes

Let $R_ K=\max _{a\in K}|a|$. For $|w|<1/(2R_ K+2)$ we have $|1-aw|\geq 1/2$, so this expression is a smooth function of $(a,w)$ on $K\times \{ |w|<1/(2R_ K+2)\} $ whose derivatives in $w$ of every order are bounded uniformly in $a\in K$; at $w=0$ it takes the value $-\log (1+|a|^2)$. Differentiating under the integral sign, we see that $q$ extends to a smooth function near infinity, and that $q+\log (1+|w|^2)=\int _ K\log |1-aw|^2\, \mathrm d\mu (a)-c$ is harmonic there, because each $\log |1-aw|^2$ is harmonic in $w$. Since $\omega _{\mathrm{FS}}=\mathrm{dd}^{\mathrm{c}}\log (1+|w|^2)$ in this chart and $\mu $ vanishes near infinity, $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q=0=\mu $ there. Hence $q$ is quasi-plurisubharmonic on $\mathbb P^1$, $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q=\mu \geq 0$ globally, and $q$ is a nonpositive $\omega _{\mathrm{FS}}$-plurisubharmonic function.

At infinity both the Lelong number and the point mass vanish, because $q$ is smooth there and $K\subset \mathbb C$. At an affine point $x$, the Lelong number of $q$ is that of the local subharmonic function $q+\rho $ (corollary 6.9.3), and by (6.7.8) and (6.7.5) it equals $\lim _{r\downarrow 0}\mu (B(x,r))=\mu (\{ x\} )$. $\square$