%Potential Theory in Several Complex Variables

\def\dr{{\cal D}}
\def\er{{\cal E}}
\def\hc{{\cal H}}
\def\mr{{\cal M}}
\def\oc{{\cal O}}

\def\div{{\rm div}}
\def\Div{{\rm Div}}
\def\psh{{\rm Psh}}
\def\reg{{\rm reg}}
\def\sing{{\rm sing}}
\def\supp{{\rm Supp}}
\def\loc{{\rm loc}}
\def\lip{{\rm lip}}
\def\gge{{\scriptscriptstyle\ge}}

\input demail.mac
\input broc.mac
\input mathghb.xmm

{\tafontt\baselineskip=18pt\parindent=0cm
Potential Theory in\vskip4pt
Several Complex Variables}
\titlearunning{ Potential Theory in Several Complex Variables}
\vskip10pt
{\noindent\hangindent1.2cm\hangafter-3{\it
Jean-Pierre Demailly\hfill\break
Universit\'e de Grenoble I, Institut Fourier, BP 74\hfill\break
URA 188 du C.N.R.S., 38402 Saint-Martin d'H\`eres, France

}}
\vskip14pt

This work is the second part of our survey article on Monge-Amp\`ere
operators. We are concerned here with the theory of complex
$n$-dimensional capacities generalizing the usual logarithmic capacity
in~$\bbbc$, and with the related notions of pluripolar and negligible
sets. Decisive progress in the theory have been made by Bedford-Taylor
[B-T1], [B-T2]. The present exposition, which is an expansion of
lectures given in Nice at the Centre International de Math\'ematiques
Pures et Appliqu\'ees (ICPAM) in 1989, borrows much to these
papers. The last section on comparison of capacities is based on the
work of Alexander and Taylor~[A-T]. We are indebted to Z.~B\l ocki and
D.~Coman for pointing out a few mistakes in the original version.
Z.~B\l ocki also suggested to derive the improved logarithmic growth
estimate (due independently to H.~El~Mir and J.~Siciak) from our proof
of Josefson's theorem on the equivalence between locally pluripolar and
globally pluripolar sets.

\titleb{10. Capacities, Regularity and Capacitability}
\titlebrunning{~\hfill~}
The goal of this section is to discuss a few fundamental notions and
results of capacity theory. The reader will find a much more complete
study in U.~Cegrell's memoir~[Ceg]. All topological spaces occurring
here are assumed to be {\it Hausdorff}.

\definition{10.1}{Let $\Omega$ be a topological space. A capacity is a set
function $\ovl c:E\mapsto \ovl c(E)$ defined on all subsets
$E\subset\Omega$ with values in $[0,+\infty]$,
satisfying the axioms {\rm(a,b,c)} below:
\smallskip
\item{\rm(a)} If $E_1\subset E_2\subset\Omega$, then 
$\ovl c(E_1)\le\ovl c(E_2)$.
\smallskip
\item{\rm(b)} If $E_1\subset E_2\subset\ldots\subset\Omega$, then
$\ovl c(\bigcup E_j)=\lim_{j\to+\infty}\ovl c(E_j)$.
\smallskip
\item{\rm(c)} If $K_1\supset K_2\supset\ldots$ are compact subsets,
then $\ovl c(\bigcap K_j)=\lim_{j\to+\infty}\ovl c(K_j)$.
\smallskip
\noindent The capacity $\ovl c$ is said to be subadditive if
moreover $\ovl c(\emptyset)=0$ and $\ovl c$ satisfies the further axiom:
\smallskip
\item{\rm(d)}
If $E_1,E_2,\ldots$ are subsets of $\Omega$, then
$\ovl c(\bigcup E_j)\le\sum_j\ovl c(E_j).$}

In our applications, we will have to consider set functions which are
only defined on the collection of Borel subsets. We thus introduce:
\titlebrunning{10. Capacities, Regularity and Capacitability}

\definition{10.2}{A precapacity is a set function $c:E\mapsto c(E)$
defined on all Borel subsets $E\subset\Omega$ with values in $[0,+\infty]$,
satisfying \hbox{axioms $10.1$~{\rm(a), (b)}.} The precapacity $c$ is
said to be inner regular if all Borel subsets satisfy
$$c(E)=\sup_{K\,{\rm compact}\,\subset E} c(K).\leqno{\rm(i)}$$
Similarly, $c$ is said to be outer regular if all Borel subsets $E$
satisfy
$$c(E)=\inf_{G\,{\rm open}\,\supset E} c(G)\leqno{\rm(o)}$$}

When $c$ is a precapacity and $E\subset\Omega$ is an arbitrary subset, the
inner capacity $c_\star(E)$ and the outer capacity $c^\star(E)$ are defined by 
$$\leqalignno{
c_\star(E)&=\sup_{K\,{\rm compact}\,\subset E} c(K),&(10.3\,{\rm i})\cr
c^\star(E)&=\inf_{G\,{\rm open}\,\supset E} c(G).&(10.3\,{\rm o})\cr}$$

\statement{(10.4) Proposition}{Let $c$ be a precapacity. If $c$ is
outer regular, then $c^\star$ is a capacity. Moreover $c^\star$ is
subadditive as soon as $c$ is subadditive.}

\proof{It is clear that $c^\star$ satisfies 10.1~(a). Moreover, for any set
$E\subset\Omega$, there is a countable intersection $\wt E=\bigcap G_\ell$
of open sets containing $E$ such that $c(\wt E)=c^\star(E)$ (take
$G_\ell\supset E$ with $c(G_\ell)<c^\star(E)+1/\ell$).
When $E_1\subset E_2\ldots$, we can arrange that
$\wt E_1\subset\wt E_2\ldots$, after replacing $\wt E_j$
by $\bigcap_{k\ge j}\wt E_k$ if necessary. Since $c$ satisfies 10.1~(b)
for Borel subsets, we conclude that $\wt E=\bigcup\wt E_j$ has
precapacity $c(\wt E)=\lim c(\wt E_j)=\lim c^\star(E_j)$, hence
by the outer regularity
$$c^\star(E)\le c^\star(\wt E)=c(\wt E)=\lim c^\star(E_j).$$
The opposite inequality is clear, thus $c^\star$ satisfies 10.1~(b).
Finally, if $K_j$ is a decreasing sequence of compact sets,
any open set $G$ containing their \hbox{intersection} contains one of the
sets~$K_j$, so~ $c(G)\ge\lim c^\star(K_j)$~ and\break
\hbox{$c^\star(\bigcap K_j)
\ge\lim c^\star(K_j)$.} The opposite inequality is again clear,
thus $c^\star$ satisfies 10.1~(c). The final assertion concerning
subadditivity is easy.\qed}

\example{10.5}{Let $\Omega$ be a separable locally compact space and let
$(\mu_\alpha)$ be a family of positive Radon measures on $\Omega$. Then
$c(E)=\sup\mu_\alpha(E)$ is a subadditive precapacity; this follows
from the standard properties of measures (countable additivity,
monotone convergence theorem). The precapacity $c$ is called the
{\it upper envelope} of the family of measures $(\mu_\alpha)$. In general,
$c$ does not satisfy the additivity property
$$E_1,E_2~{\rm disjoint}~\Rightarrow
c(E_1\cup E_2)=c(E_1)+c(E_2)\,;$$
for a specific example, consider the measures
$\mu_1=\delta_0$, $\mu_2=d\lambda$ on $\bbbr$ and the sets
$E_1=\{0\}$, $E_2={}]0,1]\,;$ then
$$c(\{0\})=1,\quad
c(]0,1])=1,\quad
c([0,1])=1.$$
Moreover, the precapacity $c=\sup\mu_\alpha$ is inner regular because all
Radon measures on a separable locally compact space are inner regular.
However, $c$ need not be outer regular: for instance, take 
$d\mu_\alpha(x)=\alpha^{-1}\rho(x/\alpha)dx$
on $\bbbr$, $\alpha>0$, where $\rho\ge0$ is a function with support
in $[-1,1]$ and $\int_\bbbr\rho(x)dx=1\,;$ then $c(\{0\})=0$ but every
neighborhood of $0$ has capacity $1$.\qed}

\definition{10.6}{Let $c$ be a precapacity on~$\Omega$. A set
$E\subset\Omega$ is said to be $c$-capacitable if
$c_\star(E)=c^\star(E)$.}

By definition, the precapacity $c$ is (inner and outer) regular
if and only if all Borel subsets are $c$-capacitable.

We are now going to prove a general capacitability theorem due to
G.~Choquet. Before doing so, we need a few results about $K$-analytic
spaces.

\definition{10.7}{Let $X$ be a topological space. Then
\medskip
\item{\rm(a)} a $F_\sigma$ subset of $X$ is a countable union of
closed subsets of $X\,;$
\smallskip
\item{\rm(b)} a $F_{\sigma\delta}$ subset of $X$ is a 
countable intersection of $F_\sigma$ subsets of $X$. 
\smallskip
\item{\rm(c)} the space $X$ is said to be a $K_\sigma$ 
$($resp.\ $K_{\sigma\delta})$ space if it is homeomorphic to some 
$F_\sigma$ $($resp.\ $F_{\sigma\delta})$ subset of a compact space
$W$.
\vskip0pt}

\statement{(10.8) Properties}{\smallskip
\item{\rm(a)}
Every closed subset $F$ of a $K_{\sigma\delta}$ space $X$ is a
$K_{\sigma\delta}$ space.
\smallskip
\item{\rm(b)}
Every countable disjoint sum $\coprod X_j$ of $K_{\sigma\delta}$ spaces is
a $K_{\sigma\delta}$ space.
\smallskip
\item{\rm(c)}
Every countable product $\prod X_j$ of $K_{\sigma\delta}$ spaces is
a $K_{\sigma\delta}$ space.}

\proof{(a) Write $X=\bigcap_{\ell\ge1}G_\ell$ and
$G_\ell=\bigcup_{m\ge1}K_{\ell m}$ where $K_{\ell m}$ are closed subsets of
a compact space $W$. If $\ovl F$ is the closure of $F$ in $W$, we have
$$F=X\cap\ovl F=\bigcap_{\ell\ge1}G_\ell\cap\ovl F,\qquad
G_\ell\cap\ovl F=\bigcup_{m\ge1}K_{\ell m}\cap\ovl F.$$
\medskip
\pritem{(b)} Let $(X_j)_{j\ge 1}$ be $K_{\sigma\delta}$ spaces and
write for each $j$
$$X_j=\bigcap_{\ell\ge1}G^j_\ell,\qquad
G^j_\ell=\bigcup_{m\ge1}K^j_{\ell m}$$
where $K^j_{\ell m}$ is a closed subspace of a compact space $W_j$, 
and let $\{\star\}$ be a one-point topological space.
Then $\coprod W_j$ can be embedded in the compact space 
$W=\prod(W_j\amalg\{\star\})$ via the obvious map
which sends $w\in W_j$ to $(\star\ld\star,w,\star,\ldots)$ with
$w$ in the $j$-th position. Now $X=\coprod X_j$ can be written
$$X=\bigcap_{\ell\ge1}G_\ell,\qquad
G_\ell=\bigcup_{m\ge1}\coprod_{j\ge1}K^j_{\ell m}.$$
As $K^j_{\ell m}$ is sent onto a closed set by the embedding $\coprod W_j
\to W$, we conclude that $X$ is a $K_{\sigma\delta}$ space.
\medskip
\pritem{(c)} With the notations of (b), write $X=\prod X_j$ as
$$\eqalign{
X=\bigcap_{\ell\ge1}G_\ell,\qquad
G_\ell=G^1_\ell\times G^2_{\ell-1}\times\ldots\times G^\ell_1
&{}\times W_{\ell+1}\times\ldots\times W_j\times\ldots,\cr
G_\ell=\bigcup_{m_1\ld m_\ell\ge1}K^1_{\ell\,m_1}\times
K^2_{\ell{-}1\, m_2}\times\ldots\times K^\ell_{1\,m_\ell}
&{}\times W_{\ell+1}\times\ldots\times  W_j\times\ldots\cr}$$
where each term in the union is closed in
$W=\prod W_j.$\qed}

\definition{10.9}{A space $E$ is said to be $K$-analytic if $E$ is a
continuous image of a $K_{\sigma\delta}$ space $X$.}

\statement{(10.10) Proposition}{Let $\Omega$ be a topological space and
let $E_1,E_2,\,\ldots$ be $K$-analytic subsets of~$\Omega$.
Then $\bigcup E_j$ and $\bigcap E_j$ are $K$-analytic.}

\proof{Let $f_j:X_j\to E_j$ be a continuous map from a 
$K_{\sigma\delta}$ space onto $E_j$. Set $X=\coprod X_j$ and
$f=\coprod f_j:X\to\Omega$.
Then $X$ is a $K_{\sigma\delta}$ space, $f$ is continuous and
$f(X)=\bigcup E_j$. Now set
$$\eqalign{
X={}&\big\{x=(x_1,x_2,\ldots)\in\prod X_j\,;~f_1(x_1)=f_2(x_2)=\cdots\big\},\cr
&f:X\to\Omega,\qquad f(x)=f_1(x_1)=f_2(x_2)=\cdots.\cr}$$
Then $X$ is closed in $\prod X_j$, so $X$ is a $K_{\sigma\delta}$
space by 10.8 (a,c) and $f(X)=\bigcap E_j.$\qed}

\corollary{10.11}{Let $\Omega$ be a separable locally compact space.
Then all Borel subsets of $\Omega$ are $K$-analytic.}

\proof{Any open or closed open set in $\Omega$ is a countable union of
compact subsets, hence $K$-analytic. On the other hand, Prop.~10.10 shows that
$${\cal A}=\{E\subset\Omega\,;~E~{\rm and}~\Omega\setminus E~{\rm
are}~K\hbox{\rm-analytic}\}$$
is a $\sigma$-algebra. Since ${\cal A}$ contains all open sets in $E$,
${\cal A}$ must also contain all Borel subsets.\qed}

Before going further, we need a simple lemma.

\lemma{10.12}{Let $E$ be a relatively compact $K$-analytic subset of a
topological space $\Omega$. There exists a compact space $T$, a
continuous map $g:T\to\Omega$ and a $F_{\sigma\delta}$ subset
$Y\subset T$ such that $g(Y)=E$.}

\proof{There is a compact space $W$, a $F_{\sigma\delta}$ subset
$X\subset W$ and a continuous map $f:X\to E$ onto $E$. Let
$$Y=\{(x,f(x))\,;\,x\in X\}\subset X\times E$$
be the graph of $f$ and $T=\ovl Y$ the closure of $Y$ in the
compact space $\ovl X\times\ovl E$. As $f$ is continuous, $Y$
is closed in $X\times\ovl E$, thus $Y=T\cap(X\times\ovl E)$.
Now, $X$ is a $F_{\sigma\delta}$ subset of $\ovl X$, so $X\times\ovl E$
is a $F_{\sigma\delta}$ subset of $\ovl X\times\ovl E$ and $Y$ is a
$F_{\sigma\delta}$ subset of $T$. Finally $E$ is the image of $Y$
by the second projection $g:T\to\ovl E.$\qed}

\statement{(10.13) Choquet's capacitability theorem}{Let $\Omega$ be a 
$K_\sigma$ space and let $\ovl c$ be a capacity on $\Omega$.
Then every $K$-analytic subset $E\subset\Omega$ satisfies
$$\ovl c(E)=\sup_{K\,{\rm compact}\,\subset E}
\ovl c(K).$$}

\proof{As $\Omega$ is an increasing union of compact sets $L_j$,
axiom~10.1~(b) implies $\ovl c(E)=\lim_{j\to+\infty}\ovl c(E\cap L_j)$;
we may therefore assume that $E$ is relatively compact in $\Omega$.
Then Lemma~10.12 shows that there is a $F_{\sigma\delta}$ subset $Y$
in a compact space $T$ and a continuous map $g:T\to\Omega$ such that
$g(Y)=E$. It is immediate to check that the set function $g^\star\ovl c$
on $T$ defined by $g^\star\ovl c(E)=\ovl c(g(E))$ is a capacity.
Hence we are reduced to proving the theorem
when $\Omega$ is a compact space and $E$ is a $F_{\sigma\delta}$ subset
of $\Omega$. We then write
$$E=\bigcap_{\ell\ge1}G_\ell,\qquad
G_\ell=\bigcup_{m\ge1}K_{\ell m}$$
where $K_{\ell m}$ is a closed subset of $\Omega$. Without loss of
generality, we can arrange that $K_{\ell m}$ is increasing in $m$. Fix
$\lambda<\ovl c(E)$. Then
$$E=G_1\cap\bigcap_{\ell\ge2}G_\ell=
\bigcup_{m\ge1}(K_{1m}\cap\bigcap_{\ell\ge2}G_\ell)$$
and axiom (b) implies that exists a subset
$E_1=K_{1m_1}\cap\bigcap_{\ell\ge2}G_\ell$
of $E$ such that $\ovl c(E_1)>\lambda$.
By induction, there is a decreasing sequence
$E\supset E_1\supset\ldots\supset E_s$ with
$$E_s=K_{1m_1}\cap\ldots\cap K_{sm_s}\cap\bigcap_{\ell\ge s+1}G_\ell$$
and $\ovl c(E_s)>\lambda$.
Set $K=\bigcap K_{sm_s}=\bigcap E_s\subset E$. Axiom 10.1~(c) implies
$$\ovl c(K)=\lim_{s\to+\infty}\ovl c
\big(K_{1m_s}\cap\ldots\cap K_{sm_s}\big)\ge
\lim_{s\to+\infty}\ovl c(E_s)\ge\lambda$$
and the theorem is proved.\qed}

\corollary{10.14}{Let $c$ be an outer regular precapacity on a
separable locally compact space~$\Omega$. Then $c$ is also inner regular
and every $K$-analytic subset of $\Omega$ is $c$-capacitable.}

\proof{By Prop.~10.4, we know that $c^\star$ is a capacity. Choquet's
theorem~10.13 implies that $c^\star(E)=\sup_{K\,{\rm compact}\,\subset E}
c^\star(K)$ for every $K$-analytic set $E$ in~$\Omega$. By the outer
regularity we have $c^\star(K)=c(K)$, hence $c^\star(E)=c_\star(E)$
and $c$ is inner regular.\qed}

\titleb{11. Monge-Amp\`ere Capacities and Quasicontinuity}
Let $\Omega$ be a bounded open subset of $\bbbc^n$. We denote by
$P(\Omega)$ the set of all plurisubharmonic functions that are $\not\equiv
-\infty$ on each connected component of $\Omega$. The following
fundamental definition has been introduced in [B-T2].

\definition{11.1}{For every Borel subset $E\subset\Omega $, we set 
$$c(E,\Omega)=\sup\Big\{\int_E(dd^cu)^n\,;~
u\in P(\Omega),~0\le u\le1\Big\}.$$}

The Chern-Levine-Nirenberg inequalities show that
$c(E,\Omega)<+\infty$ as soon as $E\compact\Omega$. If $\Omega
\subset B(z_0,R)$, we can choose $u(z)=R^{-2}|z-z_0|^2$ and we obtain
therefore
$$c(E,\Omega)\ge{2^nn!\over\pi^nR^{2n}}\,\lambda(E)\leqno(11.2)$$
where $\lambda$ is the Lebesgue measure. As a special case of Example~10.5,
we see that $c(\bu,\Omega)$ is the upper envelope of the family of
measures $\mu_u=(dd^cu)^n$, $u\in P(\Omega)$, $0\le u\le 1$.
In particular $c(\bu,\Omega)$ is a subadditive and inner regular precapacity;
it is also outer regular, but this fact is non trivial and will be proved
only in \S~14. The set function $c(\bu,\Omega)$, resp.\ $c^\star(\bu,\Omega)$,
is called the {\it relative Monge-Amp\`ere precapacity, resp.\ capacity},
of~$\Omega$. We first compare capacities associated to different open sets
$\Omega$.

\statement{(11.3) Proposition}{Let $\Omega_1\subset\Omega_2\compact\bbbc^n$. Then
\medskip
\item{\rm(a)} $c(E,\Omega_1)\ge c(E,\Omega_2)$ for all Borel subsets
$E\subset\Omega_1$.
\medskip
\item{\rm(b)} Let $\omega\compact\Omega_1$. There exists a constant $A>0$
such that for all Borel subsets $E\subset\omega$ we have
$c(E,\Omega_1)\le A\,c(E,\Omega_2)$.
\vskip0pt}

\proof{Since every plurisubharmonic function $u\in P(\Omega_2)$ with
$0\le u\le 1$ induces a plurisubharmonic function in $P(\Omega_1)$ with
the same property, (a)~is clear.
\medskip
\pritem{(b)} Use a finite covering of $\ovl\omega$ by open balls
contained in $\Omega_1$ and cut $E$ into pieces. The proof is then
reduced to the case when $\omega\compact\Omega_1$ are concentric balls,
say $\Omega_1=B(0,r)$ and $\omega=B(0,r-\varepsilon)$. For every $u\in
P(\Omega_1)$ such that $0\le u\le 1$, set
$$\wt u(z)=\cases{
\max\{u(z),\lambda(|z|^2-r^2)+2\}~ &\quad on $\;\;\Omega_1,$\cr
\lambda(|z|^2-r^2)+2~ &\quad on $\;\;\Omega_2\setminus\Omega_1.$\cr}$$
Choose $\lambda$ so large that $\lambda((r-\varepsilon)^2-r^2)\le -2$.
Then $\wt u\in P(\Omega_2)$ and $\wt u=u$ on~$\omega$. Moreover
$0\le\wt u\le M$ for some constant $M>0$, thus for $E\subset\omega$ we get
$$\int_E(dd^cu)^n=\int_E(dd^c\wt u)^n\le M^nc(E,\Omega_2).$$
Therefore $c(E,\Omega_1)\le M^nc(E,\Omega_2).$\qed}

As a consequence of Prop.~11.3, it is in general harmless to shrink the
domain $\Omega$ when capacities have to be estimated.

\statement{(11.4) Proposition}{Let $K$ be a compact subset of $\Omega$ and
$\omega\compact\Omega$ a neighborhood of $K$. There is a constant $A>0$
such that for every $v\in P(\Omega)$
$$c\big(K\cap\{v<-m\},\Omega\big)\le A\|v\|_{L^1(\ovl\omega
)}\cdot{1\over m}.$$}

\proof{For every $u\in P(\Omega),~0\le u\le 1$, Prop.~1.11 implies
$$\int_{K\cap\{v<-m\}}(dd^cu)^n\le
{1\over m}\int_K|v|(dd^cu)^n\le
{1\over m}C_{K,\ovl\omega}\|v\|_{L^1(\ovl\omega)}.\eqno{\sq}$$}

\definition{11.5}{A set $P\subset\Omega$ is said to be globally
pluripolar in $\Omega$ if there exists $v\in P(\Omega)$ such that
$P\subset\{v=-\infty\}$.}

\corollary{11.6}{If $P$ is pluripolar in $\Omega$, then
$$c^\star(P,\Omega)=0.$$}

\proof{Write $P\subset\{v=-\infty\}$ and
$\Omega=\bigcup_{j\ge1}\Omega_j$ with $\Omega_j\compact\Omega$.
Prop.~11.4 shows that there is an open set 
$G_j=\Omega_j\cap\{v<-m_j\}$ with $c(G_j,\Omega)<\varepsilon 2^{-j}$. 
Then $\{v=-\infty\}\subset G=\bigcup G_j$
and $c(G,\Omega)<\varepsilon.$\qed}

\statement{(11.7) Proposition}{Let $v_k,~v\in P(\Omega)$ be locally
bounded plurisubharmonic functions such that $(v_k)$ decreases to $v$.
Then for every compact subset\break $K\subset\Omega$ and every $\delta>0$
$$\lim_{k\to+\infty}c\big(K\cap\{v_k>v+\delta\},\Omega\big)=0.$$}

\proof{It is sufficient to show that
$$\sup_{u\in P(\Omega),0\le u\le1}\int_K(v_k-v)(dd^cu)^n$$
tends to 0, because this supremum is larger than
$\delta.c(K\cap\{v_k>v+\delta\},\Omega)$.
By cutting $K$ into pieces and modifying $v,v_k,u$ with the max
construction, we may assume that $K\subset\Omega=B(0,r)$ are concentric
balls and that all functions $v,v_k,u$ are equal to $\lambda
(|z|^2-r^2)+2)$ on the corona $\Omega\setminus\omega $, $\omega
=B(0,r-\varepsilon)$. An integration by parts yields 
$$\int_\Omega(v_k-v)(dd^cu)^n=
-\int_\Omega d(v_k-v)\wedge d^cu\wedge(dd^cu)^{n-1}.$$
The Cauchy-Schwarz inequality implies that this integral is bounded by
$$A\Big(\int_\Omega d(v_k-v)\wedge d^c(v_k-v)\wedge(dd^cu)^{n-1}\Big)^{1/2}$$
where
$$A^2=\int_\Omega du\wedge d^cu\wedge(dd^cu)^{n-1}\le
\int_\Omega dd^c(u^2)\wedge(dd^cu)^{n-1}$$
and this last integral depends only on the constants $\lambda ,r$.
Another integration by parts yields
$$\eqalign{
\int_\Omega\!d(v_k-v)\wedge d^c(v_k-v)\wedge(dd^cu)^{n-1}&=
\int_\Omega\!(v_k-v)dd^c(v-v_k)\wedge(dd^cu)^{n-1}\cr
&\le\int_\Omega\!(v_k-v)dd^cv\wedge(dd^cu)^{n-1}.\cr}$$
We have thus replaced one factor $dd^cu$ by $dd^cv$ in the integral.
Repeating the argument $(n-1)$ times we get
$$\int_\Omega(v_k-v)(dd^cu)^n\le
C\Big(\int_\Omega(v_k-v)(dd^cv)^n\Big)^{1/2^n}$$
and the last integral converges to 0 by the bounded convergence theorem.\qed}

\statement{(11.8) Theorem {\rm(quasicontinuity of plurisubharmonic
functions)}}{Let $\Omega$ be a bounded open set in $\bbbc^n$ and 
$v\in P(\Omega)$. Then for each $\varepsilon>0$, there is an open 
subset $G$ of $\Omega$ such that $c(G,\Omega)<\varepsilon$ and 
$v$ is continuous on $\Omega\setminus G$.}

\proof{Let $\omega\compact\Omega$ be arbitrary. We first show that there
exists $G\subset\omega$ such that $c(G,\Omega)<\varepsilon $ and $v$
continuous on $\omega\setminus G$. For $m>0$ large enough, the set
$G_0=\omega\cap\{v<-m\}$ has capacity $<\varepsilon/2$ by Prop.~11.4.
On $\omega\setminus G_0$ we have $v\ge -m$, thus $\wt
v=\max\{v,-m\}$ coincides with $v$ there and $\wt v$ is locally
bounded on $\Omega$. Let $(v_k)$ be a sequence of smooth plurisubharmonic functions
which decrease to $\wt v$ in a neighborhood of $\ovl\omega$.
For each $\ell\ge1$, Prop.~11.7 shows that there is an index $k(\ell)$
and an open set
$$G_{k(\ell)}=\omega\cap\{v_{k(\ell)}>\wt v+1/\ell\}$$
such that $c(G_{k(\ell)},\Omega)<\varepsilon 2^{-\ell-1}$.
Then $G=G_0\cup\bigcup G_{k(\ell)}$ has capacity $c(G,\Omega
)<\varepsilon $ by subadditivity and $(v_{k(\ell)})$ converges uniformly to
$\wt v=v$ on $\omega\setminus G$. Hence $v$ is continuous on $\omega
\setminus G$. Now, take an increasing sequence $\omega_1\subset\omega
_2\subset\ldots$ with $\bigcup\omega_j=\Omega$ and $G_j\subset\omega_j$
such that $c(G_j,\Omega)<\varepsilon 2^{-j}$ and $v$ continuous on
$\omega_j\setminus G_j$. The set $G=\bigcup G_j$ satisfies all
requirements.\qed}

As an example of application, we prove an interesting inequality for the
Monge-Amp\`ere operator.

\statement{(11.9) Proposition}{Let $u,v$ be locally bounded plurisubharmonic functions
on~$\Omega$. Then we have an inequality of measures
$$(dd^c\max\{u,v\})^n\ge
\bbbone_{\{u\ge v\}}(dd^cu)^n+\bbbone_{\{u<v\}}(dd^cv)^n.$$}

\proof{It is enough to check that
$$\int_K(dd^c\max\{u,v\})^n\ge\int_K(dd^cu)^n$$
for every compact set $K\subset\{u\ge v\}\,;$ the other term is then
obtained by reversing the roles of $u$ and $v$. By shrinking $\Omega $,
adding and multiplying with constants, we may assume that $0\le u,v\le
1$ and that $u,v$ have regularizations $u_\varepsilon=u\star\rho_\varepsilon$,
$v_\varepsilon=v\star\rho_\varepsilon$ with $0\le u_\varepsilon,~
v_\varepsilon\le 1$ on $\Omega$.
\hbox{Let $G\subset\Omega$} be an open set of small capacity such that
$u$, $v$ are continuous \hbox{on $\Omega\setminus G$.} By Dini's lemma,
$u_\varepsilon$, $v_\varepsilon $ converge uniformly to $u$, $v$ on $\Omega
\setminus G$. Hence for any $\delta>0$, we can find an arbitrarily
small neighborhood $L$ of $K$ such that $u_\varepsilon>v_\varepsilon
-\delta$ on $L\setminus G$ for $\varepsilon $ small enough. As
$(dd^cu_\varepsilon)^n$ converges weakly to $(dd^cu)^n$ on $\Omega$,
we get
$$\eqalign{
\int_K(dd^cu)^n 
&\le\liminf_{\varepsilon\to 0}\int_L(dd^cu_\varepsilon)^n\cr
&\le\liminf_{\varepsilon\to 0}\big(\int_G(dd^cu_\varepsilon)^n
+\int_{L\setminus G}(dd^cu_\varepsilon)^n\big)\cr
&\le c(G,\Omega)+\liminf_{\varepsilon\to 0}\int_{L\setminus
G}(dd^c\max\{u_\varepsilon+\delta ,v_\varepsilon\})^n.\cr}$$
Observe that $\max\{u_\varepsilon+\delta ,v_\varepsilon\}$ coincides
with $u_\varepsilon+\delta$ on a neighborhood of $L\setminus G$. By
weak convergence again, we get
$$\int_K(dd^cu)^n\le
c(G,\Omega)+\int_{L\setminus G}(dd^c\max\{u+\delta ,v\})^n.$$
By taking $L$ very close to $K$ and $c(G,\Omega)$ arbitrarily small,
this implies
$$\int_K(dd^cu)^n\le\int_K(dd^c\max\{u+\delta ,v\})^n$$
and the desired conclusion follows by letting $\delta$ tend to 0.\qed}
           
\titleb{12. Upper Envelopes and the Dirichlet Problem}
Let $(u_\alpha)$ be a family of upper semi-continuous functions on
$\Omega$ which is locally bounded from above. Then the upper envelope
$$u=\sup_\alpha u_\alpha(z)$$
need not be upper semi-continuous, so we consider its ``upper
semi-continuous regularization"
$$u^\star(z)=\lim_{\varepsilon\to0}\sup_{B(z,\varepsilon)}u\ge u(z).$$
It is easy to check that $u^\star$ is upper semi-continuous and that $u^\star$
is the smallest upper semi-continuous function $\ge u$.

Let $B(z_j,\varepsilon_j)$ be a countable basis of the topology of
$\Omega$.
For each $j$, let $(z_{jk})$ be a sequence in $B(z_j,\varepsilon_j)$
such that
$$\sup_k u(z_{jk})=\sup_{B(z_j,\varepsilon_j)}u,$$
and for each $(j,k)$, let $\alpha(j,k,\ell)$ be a sequence of indices
$\alpha$ such that $u(z_{jk})=\sup_\ell u_{\alpha(j,k,\ell)}(z_{jk})$.
Set
$$v=\sup_{j,k,\ell}u_{\alpha(j,k,\ell)}.$$
Then $v\le u$ and $v^\star\le u^\star$. On the other hand
$$\sup_{B(z_j,\varepsilon_j)}v 
\ge\sup_k v(z_{jk})\ge\sup_{k,\ell}u_{\alpha(j,k,\ell)}(z_{jk})
=\sup_k u(z_{jk})=\sup_{B(z_j,\varepsilon_j)}u.$$
As every ball $B(z,\varepsilon)$ is a union of balls $B(z_j,\varepsilon_j)$,
we easily conclude that $v^\star\ge u^\star$, hence $v^\star=u^\star$. 
Therefore:

\statement{(12.1) Choquet's lemma}{Every family $(u_\alpha)$ has a 
countable subfamily  $(u_{\alpha(j)})$ whose upper envelope $v$
satisfies $v\le u\le u^\star=v^\star$.}

\statement{(12.2) Proposition}{If all $u_\alpha$ are plurisubharmonic, then $u^\star$ 
is plurisubharmonic and equal almost everywhere to $u$.}

\proof{By Choquet's lemma we may assume that $(u_\alpha)$ is
countable. Then $u=\sup u_\alpha$ is a Borel function. For every
$(z_0,a)\in\Omega\times\bbbc^n$, $u_\alpha$ satisfies the mean value
inequality on circles, hence
$$u(z_0)=\sup u_\alpha(z_0)\le 
\sup\int^{2\pi}_0u_\alpha(z_0+ae^{\ii\theta}){d\theta\over 2\pi}
\le\int^{2\pi}_0u(z_0+ae^{\ii\theta}){d\theta\over 2\pi}.$$
It follows easily that each convolution $u\star\rho_\varepsilon $ also
satisfies the mean value inequality, thus $u\star\rho_\varepsilon $ is
smooth and plurisubharmonic.
Therefore $(u\star\rho_\varepsilon)\star\rho_\eta$ is increasing in $\eta$.
Letting $\varepsilon $ tends to 0, we see that $u\star\rho_\eta$ in
increasing in $\eta$. Since $u\star\rho_\varepsilon $ is smooth and
$u\star\rho_\varepsilon\ge u$ by the mean value inequality, we also have
$u\star\rho_\varepsilon\ge u^\star$. By the upper semi-continuity we get
$\lim_{\varepsilon\to 0}u\star\rho_\varepsilon=u^\star$, in particular
$u^\star $ is plurisubharmonic and coincides almost everywhere with the $L^1_{\rm loc}$
limit $u.$\qed}

In the sequel, we need the fundamental result of Bedford-Taylor\break
[B-T1] on the solution of the Dirichlet problem for complex Monge-Amp\`ere
equations.

\theorem{12.3}{Let $\Omega\compact\bbbc^n$ be a smooth strongly
pseudoconvex domain and let $f\in C^0(\partial\Omega)$ be a continuous
function on the boundary. Then
$$u(z)=\sup\{v(z)\,;~v\in P(\Omega)\cap C^0(\ovl\Omega),~v\le
f~{\rm on}~\partial\Omega\}$$
is continuous on $\ovl\Omega$ and plurisubharmonic on $\Omega $, and solves
the Dirichlet problem
$$(dd^cu)^n=0~{\rm on}~\Omega ,\quad u=f~{\rm on}~\partial\Omega.$$}

\proof{The main difficulty is to obtain sufficient regularity of $u$ when
$f$ is smooth, so as to be able to analyze the local convexity of $u$
at any point. The proof given in [B-T1] consists of three steps.
\medskip
\noindent{\it Step 1. The upper envelope $u$ is continuous on $\ovl\Omega$
and $u=f$ on $\partial\Omega$.}\smallskip
Let $g\in C^2(\ovl\Omega)$ be an approximate extension of $f$ such that
$|g-f|<\varepsilon $ on $\partial\Omega$ and let $\psi<0$ be a smooth
strongly plurisubharmonic exhaustion of $\Omega$. Then
$g-\varepsilon+A\psi $ is plurisubharmonic for $A>0$ large enough and
$g-\varepsilon+A\psi=g-\varepsilon\le f$ on $\partial\Omega $, hence
$g-\varepsilon+A\psi\le u$ on $\ovl\Omega$.  Similarly, for all $v\in
P(\Omega)\cap C^0(\ovl\Omega)$ with $v\le f$ on $\partial\Omega $, the
function $v-g-\varepsilon+A\psi $ equals $v-g-\varepsilon\le 0$ on
$\partial\Omega $ and is plurisubharmonic for $A$ large, thus
$v-g-\varepsilon+A\psi\le 0$ on $\ovl\Omega$ by the maximum principle.
Therefore we get $u\le g+\varepsilon -A\psi \,;$ as $\varepsilon $
tends to 0, we see that $u=f$ on $\partial\Omega$ and that $u$ is
continuous at every point of $\partial\Omega$. Since
$g+\varepsilon+A\psi=g+\varepsilon>f$ on  $\partial\Omega$, there
exists $\delta>0$ such that $u^\star<g+\varepsilon +A\psi $ on
$\Omega\setminus\Omega_\delta $, where 
$\Omega_\delta=\{\psi<-\delta\}$. For  $\eta>0$ small enough, the
regularizations of $u^\star$ satisfy
$u^\star\star\rho_\eta<g+\varepsilon+A\psi $ on a neighborhood
of~$\partial\Omega_\delta$. Then we let
$$v_\varepsilon=\cases{
\max\{u^\star\star\rho_\eta -2\varepsilon ,g-\varepsilon+A\psi\}&\quad on
$\;\;\Omega_\delta$\cr
g-\varepsilon+A\psi  &\quad on $\;\;\ovl\Omega\setminus\Omega
_\delta.$\cr}$$
It is clear that $v_\varepsilon $ is plurisubharmonic and continuous on 
$\ovl\Omega$ and we have\break $v_\varepsilon=g-\varepsilon\le f$ on
$\partial\Omega$, thus $v_\varepsilon\le u$ on $\ovl\Omega$. We get
therefore $u^\star\star\rho_\eta\le u+2\varepsilon $ on $\Omega_\delta$.
As $u\le u^\star\le u^\star\star\rho_\eta $, we see that
$u^\star\star\rho_\eta$ converges uniformly to $u$ on every compact
subset of $\Omega$. Hence $u$ is plurisubharmonic and continuous
on~$\ovl\Omega$.
\medskip
\noindent{\it Step 2. If $\Omega=B=B(0,1)\subset\bbbc^n$ and
$f\in C^{1+\lip}(\partial B)$, then $u\in C^{1+{\rm lip}}(B)$.}
\smallskip
Here $C^{1+\lip}$ denotes the space of functions admitting
Lipschitz continuous first derivatives; such functions have
locally bounded second derivatives almost everywhere by the
Lebesgue differentiability theorem. Since we are
going to obtain uniform estimates in terms of $||f||_{C^2(\partial B)}$,
it is enough to consider the case when $f\in C^2(\partial B)$, thanks to
a regularization argument. The $C^0$~estimate $||u||_{C^0(\ovl B)}\le
||f||_{C^0(\partial B)}$ is clear by the maximum principle. Now,
there is a $C^2$ extension $\wh f$ of $f$ to $2\ovl B$ such that
$||\wh f||_{C^2(2\ovl B)}\le C||f||_{C^2(\partial B)}$. After
adding or subtracting a sufficiently large multiple of $1-|z|^2$ to $\wh f$,
we get a plurisubharmonic extension $f'$ of $f$ to $2\ovl B$ and a
plurisuperharmonic extension $f''$ satisfying similar estimates.
Then $f'\le u\le f''$ on~$B$, the se\-cond equality being a consequence
of the maximum principle applied to the plurisubharmonic function
\hbox{$u-f''$.} We get a plurisubharmonic extension $\wh u$ of $u$ to
$2\ovl B$ by setting $\wh u=u$ on $B$ and $\wh u=f'$ on~$2\ovl B\ssm B$. Then
$\wh u\le\max\{f',f''\}$ on $2\ovl B$ and for all $z\in\partial B$,
$|h|<1$ we get 
$$\eqalign{
\wh u(z+h)&\le f(z)+\max\big\{||f'||_{C^1(2\ovl B)},
||f''||_{C^1(2\ovl B)}\big\}\,|h|\cr
&\le f(z)+C'||f||_{C^2(\partial B)}|h|.\cr}$$
The definition of $u$ as an upper envelope implies
$$\wh u(z+h)-C'||f||_{C^2(\partial B)}|h|\le u(z)~~~~\hbox{\rm on}~~B.$$
By changing $h$ into $-h$, we conclude that
$|u(z+h)-u(z)|\le C'||f||_{C^2(\partial B)}|h|$ for $z\in B$ and $|h|$ small,
thus $||u||_{C^1(\ovl B)}\le C'||f||_{C^2(\partial B)}$.

 In order to get the estimate for second derivatives, the idea is to
move $u$ and $f$ by automorphisms of~$B$ instead of ordinary translations
(this is the reason why we have to assume $\Omega=B$). Let $a=\lambda\zeta
\in B$ with $\zeta\in\partial B$ and $\lambda\in\bbbr$, $|\lambda|<1$.
We define an analytic map $\Phi_a$ on $B$ by
$$\eqalign{
\Phi_a(z)&={P_a(z)-a+(1-|a|^2)^{1/2}Q_a(z)\over 1-\langle z,a\rangle}\cr
&={(\langle z,\zeta\rangle-\lambda)\zeta+(1-\lambda^2)^{1/2}
(z-\langle z,\zeta\rangle\zeta)\over 1-\lambda\langle z,\zeta\rangle},\cr}$$
where $\langle\bu,\bu\rangle$ denotes the usual hermitian inner product
on~$\bbbc^n$, $P_a$ the orthogonal projection of $\bbbc^n$ onto
$\bbbc\{a\}$ and $Q_a=\id-P_a$. Easy computations show that
$$1-|\Phi_a(z)|^2={(1-\lambda^2)(1-|z|^2)\over
|1-\lambda\langle z,\zeta\rangle|^2},$$
from which it follows that $\Phi_a$ is an automorphism of $B$ (see~e.g.~[Rud]).
Moreover
$$\Phi_a(z)={z-a+O(|a|^2)\over 1-\langle z,a\rangle}=z-a+\langle z,a\rangle z
+O(|a|^2)$$
where $O(|a|^2)$ is uniform with respect to $(z,\zeta)\in\ovl B\times\partial B$
when $|a|$ tends to~$0$. Then $v=u\circ\Phi_a+u\circ\Phi_{-a}$
is plurisubharmonic on $B$ and
$$v=f\circ\Phi_a+f\circ\Phi_{-a}\le 2f+K|a|^2~~~~\hbox{\rm on}~~\partial B$$
where $K={\rm const}\,||f||_{C^2(\partial B)}$. As above, we conclude
that ${1\over 2}(v-K|a|^2)\le u$ on $B$, hence with $h=a-\langle z,a\rangle z$
we get
$$u(z-h)+u(z+h)\le v(z)+C'||u||_{C^1(\ovl B)}|a|^2\le 2u(z)+
C''||f||_{C^2(\partial B)}|a|^2.$$
The inverse linear map $h\mapsto a$ has norm $\le (1-|z|^2)^{-1}$, thus
we finally get
$$u(z-h)+u(z+h)-2u(z)\le C''(1-|z|^2)^{-2}||f||_{C^2(\partial B)}|h|^2.$$
By taking a convolution with a regularizing kernel $\rho_\varepsilon$
we infer
$$u_\varepsilon(z-h)+u_\varepsilon(z+h)-2u_\varepsilon(z)\le
C''\big(1-(|z|+\varepsilon)^2\big)^{-2}||f||_{C^2(\partial B)}|h|^2$$
with $u_\varepsilon=u\star\rho_\varepsilon$. A Taylor expansion of
degree two of $u_\varepsilon$ at $z$ gives
$$D^2u_\varepsilon(z)\cdot h^2\le
C''\big(1-(|z|+\varepsilon)^2\big)^{-2}||f||_{C^2(\partial B)}|h|^2.$$
As $u_\varepsilon$ is plurisubharmonic, $u_\varepsilon$ has a semi-positive
complex Hessian, that is,
\hbox{$D^2u_\varepsilon(z)\cdot h^2+D^2u_\varepsilon(z)\cdot(ih)^2\ge 0$}.
This implies
$$D^2u_\varepsilon(z)\cdot h^2\ge-D^2u_\varepsilon(z)\cdot(ih)^2\ge
-C''\big(1-(|z|+\varepsilon)^2\big)^{-2}||f||_{C^2(\partial B)}|h|^2,$$
thus $|D^2u_\varepsilon(z)|\le C''\big(1-(|z|+\varepsilon)^2\big)^{-2}
||f||_{C^2(\partial B)}$. By taking the limit as $\varepsilon$ tends to~$0$,
we infer that the distribution $D^2u$ has a $L^\infty_\loc$ density such that
$|D^2u(z)|\le C''(1-|z|^2)^{-2}||f||_{C^2(\partial B)}$, in particular
$u\in C^{1+\lip}(B)$. By exercising a little more care in the
estimates, one can show in fact that $|D^2u(z)|\le
C(1-|z|^2)^{-1}||f||_{C^2(\partial B)}$ (see A.~Dufresnoy [Duf]).
\medskip
\noindent{\it Step 3. The upper envelope $u$ satisfies $(dd^cu)^n=0$
on $\Omega$.}\smallskip
We first prove the result under the additional assumption
\hbox{$u\in C^{1+\lip}(\Omega)$,} which we know to be true if $\Omega=B$
and $f\in C^{1+\lip}(\partial\Omega)$. Then $D^2u$ has se\-cond partial
derivatives almost everywhere. As~$D^2u_\varepsilon$ converges to
$D^2u$ almost everywhere by Lebesgue's theorem, it is immediately
seen by Th.~1.7~(b) that the Monge-Amp\`ere current $(dd^cu)^n$ defined
in~\S~1 coincides with the corresponding $L^\infty_\loc$ form of type $(n,n)$
obtained by a pointwise computation. The plurisubharmonicity of $u$ implies
$\det(\partial^2u/\partial z_j\partial\ovl z_k)\ge0$.
If the determinant were not equal to $0$ almost everywhere, there would
exist a point $z_0\in\Omega$, say $z_0=0$ for simplicity, at which $u$
would have second derivatives and such that
$\det(\partial^2u/\partial z_j\partial\ovl z_k(0))>0$.
Then the Taylor expansion of $u$ at $z_0$ would give
$$u(z)=\re P(z)+\sum c_{jk}z_j\ovl z_k+o(|z|^2)$$
where $P$ is a holomorphic polynomial of degree 2 and $(c_{jk})$ is a
positive definite hermitian matrix. Hence we would have
$u>\re P+\varepsilon $ on a small sphere $S(0,r)$ with
$\ovl B(0,r)\subset\Omega$. The function
$$v=\cases{
\max\{u,\re P+\varepsilon\} &\quad on $\;\; B(0,r)$\cr
u &\quad on $\;\;\ovl\Omega\setminus B(0,r)$\cr}$$
is then continuous on $\ovl\Omega$ and plurisubharmonic, and satisfies
$$v=u\le f~~{\rm on}~\partial\Omega.$$
By the definition of $u$, we thus have $u\ge v$ on $\ovl\Omega$.
This is a contradiction because
$$v(0)>\re P(0)=u(0).$$
Therefore we must have $(dd^cu)^n=0$, as desired.

We first get rid of the additional assumption $u\in C^{1+\lip}(\Omega)$
when \hbox{$\Omega=B$.} Supposing only $f\in C^0(\partial B)$, we select a
decreasing of functions \hbox{$f_\nu\in C^2(\partial B)$} converging to~$f$.
It is then easy to see that the sequence of associated envelopes
$u_\nu$ is decreasing and converges uniformly to~$u$, with
$||u_\nu-u||_{C^0(B)}\le||f_\nu-f||_{C^0(B)}$. Theorem~1.7 implies
therefore $(dd^cu)^n=0$. Before finishing the proof for arbitrary
strongly pseudoconvex domains~$\Omega$, we infer the following:}

\corollary{12.5}{Fix a ball $\ovl B(z_0,r)\subset\Omega$ and let
$g\in P(\Omega)$ be locally bounded. There exists a function $\wt
g\in P(\Omega)$ such that $\wt g\ge g$ on $\Omega $, $\wt g=g$ on
$\Omega\setminus B(0,r)$ and $(dd^c\wt g)^n=0$ on $B(z_0,r)$.
Moreover, for $g_1\le g_2$ we have $\wt g_1\le\wt g_2$.}

\proof{Assume first that $g\in C^0(\Omega)$. By Th.~12.3 applied on
$B(z_0,r)$, there exists a function $u$, plurisubharmonic and continuous on $\ovl
B(z_0,r)$, with $u=g$ on $S(z_0,r)$ and $(dd^cu)^n=0$ on $B(z_0,r)$. Set
$$\wt g=\cases{
u &\quad on $\;\; B(z_0,r)$\cr
g &\quad on $\;\;\Omega\setminus B(z_0,r)$.\cr}$$
By definition of $u$, we have $\wt g=u\ge g$ on $B(z_0,r)$. Moreover,
$\wt g$ is the decreasing limit of the plurisubharmonic functions 
$$g_k=\cases{
\max\{u,g+{1\over k}\} &\quad on $\;\; B(z_0,r)$\cr
g+{1\over k} &\quad near $\;\;\Omega\setminus B(z_0,r)$\cr}$$
hence $\wt g$ is plurisubharmonic. Also clearly, for $g_1\le g_2$ we have $u_1\le
u_2$, hence $\wt g_1\le\wt g_2$. For an arbitrary locally bounded
function $g\in P(\Omega)$, write $g$ as a decreasing limit of smooth
plurisubharmonic functions $g_k=g\star\rho_{1/k}$ and set $\wt
g=\lim_{k\to+\infty}\downarrow\wt g_k$. Then $\wt g$ has all
required properties.\qed}

\pproof{End of proof of Step 3 in Theorem 12.3}{Apply Cor.~12.5 to
$g=u$ on an arbitrary ball $\ovl B(z_0,r)\subset\Omega$. Then we
get a continuous plurisubharmonic function $\wt u\ge u$ with the same
boundary values as $u$ on $\partial\Omega$, so we must have~$\wt u=u$.
In particular $(dd^cu)^n=(dd^c\wt u)^n=0$ on~$B(z_0,r)$.\qed}

\titleb{13. Extremal Functions and Negligible Sets}
To study further properties of complex potential theory, it is necessary
to make a much deeper study of upper envelopes.

\definition{13.1}{A negligible set in an open set $\Omega\subset\bbbc^n$ is a
set of the form
$$N=\{z\in\Omega\,;~u(z)<u^\star(z)\}$$
where $u$ is the upper envelope of a family $(u_\alpha)$ of
plurisubharmonic functions which is locally bounded from above on~$\Omega$,
and where $u^\star$ is the upper semicontinuous regularization of~$u$.}

\statement{(13.2) Proposition}{If $\Omega\subset\bbbc^n$ is pseudoconvex,
every pluripolar set $P=\{v=-\infty\}$ in $\Omega$ is negligible.}

\proof{Let $w\in P(\Omega)\cap\ci(\Omega)$ be such that $w\ge v$
and let $u_\alpha=(1-\alpha)v+\alpha w$, $\alpha\in{}]0,1[$. Then
$u_\alpha$ is increasing in $\alpha$ and $u=\sup_\alpha u_\alpha$
satisfies
$$\matrix{
&u=-\infty\quad\hfill &{\rm on}\;\;\{v=-\infty\},\hfill\cr
&u=w\quad\hfill &{\rm on}\;\;\{v>-\infty\}.\hfill\cr}$$
Hence $u^\star=w$ and $\{u<u^\star\}=\{v=-\infty\}$.\qed}

Next we consider the extremal function associated to a subset $E$ of $\Omega:$
$$u_E(z)=\sup\{v(z);v\in P(\Omega),~v\le-1~{\rm on}~E,~v\le0~{\rm
on}~\Omega\}.\leqno(13.3)$$
Proposition 12.2 implies $u^\star_E\in P(\Omega)$ and $-1\le u^\star_E\le 0$.
We prove the following three fundamental results by a simultaneous
induction on $n$.

\statement{(13.4) Proposition}{Let $u,u_j\in P(\Omega)$ be locally bounded
functions such that $u_j$ increases to $u$ almost everywhere. Then the
measure $(dd^cu_j)^n$ converges weakly to $(dd^cu)^n$ on $\Omega$.}

\statement{(13.5) Proposition}{Let $\Omega$ be a strongly pseudoconvex 
smooth open set in~$\bbbc^n$. If $K\subset\Omega$ is compact, then 
\medskip
\item{\rm(a)}~~ $(dd^cu^\star_K)^n=0$ on $\Omega\setminus K$.
\medskip
\item{\rm(b)}~~ $c(K,\Omega)=\int_K(dd^cu^\star_K)^n=\int_\Omega 
(dd^cu^\star_K)^n$.}

\statement{(13.6) Proposition}{If a Borel set $N\subset\Omega$ is negligible,
then $c(N,\Omega)=0.$}

The proof is made in three inductive steps.
\medskip
\pritem
{\bf Step 1}: (13.4) in $\bbbc^n\Rightarrow$ (13.5) in $\bbbc^n$.\hfill\break
{\bf Step 2}: (13.5) in $\bbbc^n\Rightarrow$ (13.6) in $\bbbc^n.$\hfill\break
{\bf Step 3}: (13.4) and (13.6) in $\bbbc^n\Rightarrow$ (13.4) in $\bbbc^{n+1}$.
\medskip
\pritem{}In the case $n=1$, Prop.~13.4 is a well-known fact of
distribution theory: $u_j$ converges to $u$ in $L^1_{\rm loc}(\Omega
)$, thus $dd^cu_j$ converges weakly to $dd^cu$. By the inductive
argument, Prop.~13.4, 13.5, 13.6 hold in all dimensions.

\pproof{Proof of Step 1}{By Choquet's lemma, there is a sequence of
functions \hbox{$v_j\in P(\Omega)$} such that $v_j\le0$ on $\Omega $,
$v_j\le -1$ on $K$ and $v^\star=u^\star_K$. If we replace $v_j$ by
$\max\{-1,v_1,\ldots,v_j\}$, we see that we may assume $v_j\ge-1$ for
all $j$ and $v_j$ increasing. Then fix an arbitrary ball
$B(z_0,r)\subset\Omega\setminus K$ and consider the increasing sequence
$\wt v_j$ given by Cor.~12.5. We still have $\wt v_j\le 0$ on
$\Omega$ and $\wt v_j\le -1$ on $K$, thus $v_j\le\wt v_j\le u_K$
and $\wt v=\lim\wt v_j$ satisfies $v^\star=\wt v^\star=u^\star_K$, in
particular $\lim\wt v_j=\lim v_j=u^\star_K$ almost everywhere. Since
$(dd^c\wt v_j)^n=0$ on $B(z_0,r)$, we conclude by 13.4 that
$(dd^cu^\star_K)^n=0$ on $B(z_0,r)$ and 13.5 (a) is proved.

To prove 13.5 (b), we first observe that $-1\le u^\star_K\le 0$ on $\Omega$,
hence \hbox{$c(K,\Omega)\ge\int_K(dd^cu^\star_K)^n$} by definition of the
capacity. If $\psi<0$ is a smooth strictly plurisubharmonic exhaustion function of
$\Omega $, we have $A\psi\le -1$ on $K$ for $A$ large enough.
We can clearly assume $v_j\ge A\psi $ on $\Omega \,;$ otherwise replace
$v_j$ by $\max\{v_j,A\psi\}$. Now, let $w\in P(\Omega)$ be such that
$0\le w\le 1$ and set
$$w'=(1-\varepsilon)w-1+\varepsilon/2,~~~~
w_j=\max\{w',v_j\}.$$
Since $-1+\varepsilon/2\le w'\le -\varepsilon/2$
on $\Omega $, we have
$w_j=v_j$ as soon as $A\psi>-\varepsilon/2$, whereas
$w_j=w'\ge -1+\varepsilon/2>v_j$ on a
neighborhood of $K$. Hence for $\delta>0$ small enough Stokes' theorem
implies
$$\int_{\Omega_\delta}(dd^cv_j)^n=\int_{\Omega_\delta}(dd^cw_j)^n
\ge\int_K(dd^cw_j)^n=(1-\varepsilon)^n\int_K(dd^cw)^n.$$
By 13.4 $(dd^cv_j)^n$ converges weakly to $(dd^cu^\star_K)^n$ and we get 
$$\limsup_{j\to+\infty}\int_{\Omega_\delta}(dd^cv_j)^n\le
\int_{\ovl\Omega_\delta}
(dd^cu^\star_K)^n=\int_K(dd^cu^\star_K)^n.$$
Therefore $\int_K(dd^cw)^n\le\int_K(dd^cu^\star_K)^n$ and $c(K,\Omega
)\le\int_K(dd^cu^\star_K)^n.$\qed}

\pproof{Proof of Step 2}{Let $N=\{v<v^\star\}$ with $v=\sup v_\alpha$. By
Choquet's lemma, we may assume that $v_\alpha$ is an increasing
sequence of plurisubharmonic functions. The theorem of quasicontinuity shows that
there exists an open set $G\subset\Omega$ such that all functions
$v_\alpha$ and $v^\star$ are continuous on $\Omega\setminus G$ and
$c(G,\Omega)<\varepsilon$. Write
$$N\subset G\cup(N\cap(\Omega\setminus G))=G\cup\bigcup_{\delta
,\lambda ,\mu\in\bbbq}K_{\delta\lambda_\mu}$$
where $\delta>0$, $~\lambda<\mu $ and
$$K_{\delta\lambda\mu}=\big\{z\in\ovl\Omega_\delta\setminus
G\,;~ v(z)\le\lambda<\mu\le v^\star(z)\big\}.$$
As $v^\star$ is continuous and $v$ lower semi-continuous on $\Omega
\setminus G$, we see that $K_{\delta\lambda\mu}$ is compact. We only
have to prove that $c(K_{\delta\lambda\mu},\Omega)=0$. Set
$K=K_{\delta\lambda\mu}$ for simplicity and take an open set $\omega
\compact\Omega$. By subtracting a large constant, we may assume
$v^\star\le 0$ on $\ovl\omega$.

Multiplying by another constant, we may set $\lambda=-1$. Then all
$v_\alpha$ satisfy  $v_\alpha\le 0$ on $\omega$ and $v_\alpha\le v\le
-1$ on $K$.
We infer that the extremal function $u_K$ on $\omega$ satisfies $u_K\ge
v$, $u^\star_K\ge v^\star$, in particular $u^\star_K\ge\mu>-1$ on $K$. By
Prop.~11.9 we obtain
$$c(K,\omega)=\int_K(dd^cu^\star_K)^n\le
\int_K(dd^c\max\{u^\star_K,\mu\})^n\le
|\mu |^nc(K,\omega)$$
because $-1\le|\mu |^{-1}\max\{u^\star_K,\mu\}\le0$.
As $|\mu |<1$, we infer that $c(K,\omega)=0$, hence $c(K,\Omega)=0.$\qed}
\eject
\pproof{Proof of Step 3}{We have to show that if $\Omega\subset\bbbc^{n+1}$, 
$$\lim_{j\to+\infty}\int_\Omega\chi(dd^cu_j)^{n+1}=\int_\Omega\chi
(dd^cu)^{n+1}$$
for all test functions $\chi\in C^\infty_0(\Omega)$. That is,
$$\lim_{j\to+\infty}\int_\Omega u_j(dd^cu_j)^n\wedge\gamma
=\int_\Omega u(dd^cu)^n\wedge\gamma\leqno(13.7)$$
with $\gamma=dd^c\chi$. As all $(1,1)$-forms $\gamma$ can be written
as linear combinations of forms of the type $\ii\alpha\wedge\ovl
\alpha $, $\alpha\in\Lambda^{1,0}(\bbbc^n)^\star$, it is sufficient,
after a change of coordinates, to consider forms of the type 
$\gamma={\ii\over 2}\chi(z)dz_{n+1}\wedge d\ovl z_{n+1}$ with
$\chi\in C^\infty_0(\Omega)$.
In this case, for any locally bounded plurisubharmonic function $v$ on $\Omega $, the
Fubini theorem yields
$$\int_\Omega v(dd^cv)^n\wedge\gamma=
\int_\bbbc d\lambda(z_{n+1})\int_{\Omega(z_{n+1})}
\chi(\bu,z_{n+1})(dd^cv(\bu,z_{n+1}))^n$$
where $\Omega(z_{n+1})=\{z\in\bbbc^n;(z,z_{n+1})\in\Omega\}$
and $f(\bu,z_{n+1})$ denotes the function $z\mapsto f(z,z_{n+1})$ on
$\Omega(z_{n+1})$.
Indeed, the result is clearly true if $v$ is smooth. The general case
follows by taking smooth plurisubharmonic functions $v_j$ decreasing to $v$. The
convergence of both terms in the equality is guaranteed by Th.~1.7~(a),
combined with 1.3 and the bounded convergence theorem for the right
hand side.

In order to prove (13.7), we thus have to show
$$\lim_{j\to+\infty}\int_\omega\chi u_j(dd^cu_j)^n=\int_\omega\chi 
u(dd^cu)^n\leqno(13.8)$$
for $\omega\subset\bbbc^n$, $\chi\in C^\infty_0(\omega)$ and 
$u_j\in P(\omega)\cap L^\infty_{\rm loc}(\omega)$ increasing
to $u\in P(\omega)$ almost everywhere. To prove (13.8), we can clearly
assume $0\le\chi\le 1$ and $0\le u_j\le u\le 1$ on $\Omega$. By our
inductive hypothesis 13.4, $(dd^cu_j)^n$ converges weakly to $(dd^cu)^n$.
As $u_j\le u\le u_\varepsilon=u\star\rho_\varepsilon $, we get
$$\eqalign{
\limsup_{j\to+\infty}\int_\omega\chi u_j(dd^cu_j)^n 
&\le\lim_{\varepsilon\to 0}\lim_{j\to+\infty}\int_\omega\chi 
u_\varepsilon(dd^cu_j)^n\cr
&=\lim_{\varepsilon\to0}\int_\omega\chi u_\varepsilon
(dd^cu)^n=\int_\omega\chi  u(dd^cu)^n.\cr}$$
To prove the other inequality, let $\varepsilon>0$ and choose an open
set $G\subset\omega$ such that $c(G,\omega)<\varepsilon $ and $u,u_j$
are all continuous on $\omega\setminus G$. Let $v=\sup u_j$. Then
$v^\star=u$ because $v^\star$ and $u$ are plurisubharmonic and coincide
almost everywhere. Let $\wt u_j$ be a continuous extension of
$u_j|_{\omega\setminus G}$ to $\omega$ such that
\hbox{$0\le\wt u_j\le 1$.} For $j\ge k$ we have $u_j\ge u_k$, hence

$$\eqalign{
\int_\omega\chi u_j(dd^cu_j)^n 
&\ge\int_{\omega\setminus G}\chi\wt u_k(dd^cu_j)^n\cr
&\ge\int_\omega\chi\wt u_k(dd^cu_j)^n-\int_G(dd^cu_j)^n.\cr}$$
The last integral on the right is
$\le c(G,\omega)<\varepsilon$. Taking the limit as $j$ tends to
$+\infty$, we obtain
$$\eqalign{
\liminf_{j\to+\infty}\int_\omega\chi  u_j(dd^cu_j)^n
&\ge\int_\omega\chi\wt u_k(dd^cu)^n-\varepsilon\cr
&\ge\int_\omega\chi u_k(dd^cu)^n-2\varepsilon.\cr}$$
The second term $\varepsilon $ comes from
$\int_G(dd^cu)^n\le c(G,\omega)<\varepsilon$.
Now let $k\to+\infty$ and $\varepsilon\to 0$ to get
$$\liminf_{j\to+\infty}\int_\omega\chi  u_j(dd^cu_j)^n\ge
\int_\omega\chi  v(dd^cu)^n.$$
Moreover, the Borel set $N=\{v<u=v^\star\}$ is negligible and the inductive
hypothesis 13.6 implies $c(N,\omega)=0$. Therefore 
$$\int_\omega\chi(u-v)(dd^cu)^n\le\int_N(dd^cu)^n=0$$
and the proof is complete.\qed}

\theorem{13.9}{For each $j=1\ld q$, let $u^k_j$ be an increasing
sequence of locally bounded plurisubharmonic functions such that $u^k_j$ converges
almost everywhere to $u_j\in P(\Omega)$. Then 
\medskip
\item{\rm(a)} $dd^cu^k_1\wedge\ldots\wedge dd^cu^k_q\to
dd^cu_1\wedge\ldots\wedge dd^cu_q$
weakly.
\medskip
\item{\rm(b)} $u^k_1dd^cu^k_2\wedge\ldots\wedge dd^cu^k_q\to
u_1dd^cu_2\wedge\ldots\wedge dd^cu_q$ weakly.}

\proof{(a) Without loss of generality, we may assume $q=n$, otherwise we
complete with additional stationary sequences
$u^k_{q+1}=u_{q+1}\ld u^k_n=u_n$
where
$u_{q+1}\ld u_n$ are chosen arbitrarily in
$P(\Omega)\cap C^\infty(\Omega)$. Now apply Prop.~13.4 to 
$u_k=\lambda_1u^k_1+\cdots+\lambda_nu^k_n$,
$\lambda_j>0$, and consider the coefficient of 
$\lambda_1\ldots\lambda_n$ in $(dd^cu_k)^n$.
\medskip
\pritem{(b)} Same proof as for (13.8).\qed}

\example{13.10}{Let $\Omega=B(0,R)$ and $K=\ovl B(0,r)$. The
corresponding extremal function $u_K$ is 
$$u_K(z)=\Big(\log{R\over r}\Big)^{-1}\max
\Big\{\log{|z|\over R},\log{r\over R}\Big\}.$$
In fact, for any $v\in P(\Omega)$ with $v\le-1$ on $K$ and $v\ge 0$
on~$\Omega$, the convexity of $\log\rho\mapsto\sup_{B(0,\rho)}v$ shows
that $v\le u_K$. Formula 13.5 (b) then gives
$$c(K,\Omega)=\int_\Omega(dd^cu_K)^n=\Big(\log{R\over r}\Big)^{-n}$$
where $(dd^cu_K)^n$ is a unitary invariant measure supported on the
sphere $S(0,r)$ (see example (4.3)).\qed}

Next we quote a few elementary properties of the extremal functions
$u^\star_E$ for arbitrary sets $E\subset\Omega$.

\statement{(13.11) Properties}{\smallskip
\item{\rm(a)}
if $E_1\subset E_2\subset\Omega $, then $u^\star_{E_1}\ge u^\star_{E_2}$.
\medskip
\item{\rm(b)}
if $E\subset\Omega_1\subset\Omega_2$, then $u^\star_{E,\Omega_1}\ge
u^\star_{E,\Omega_2}$.
\medskip
\item{\rm(c)}
if $E\subset\Omega $, then $u^\star_E=u_E=-1$ on $E^0$ and
$(dd^cu^\star_E)^n=0$ on $\Omega\setminus\ovl E\,;$
hence $(dd^cu^\star_E)^n$ is supported by $\partial E$.
\medskip
\item{\rm(d)}
one has $u^\star_E\equiv 0$ if and only if there exists $v\in P(\Omega)$,
$v\le 0$ such that\break $E\subset\{v=-\infty\}$.
\medskip
\item{\rm(e)}
if $E\compact\Omega$ and if $\Omega$ is strongly pseudoconvex
with exhaustion $\psi<0$, then $u^\star_E\ge A\psi $ for some $A>0$.
\vskip0pt}

\proof{(a), (b) are obvious from Def.~(13.3); (e) is true as soon as
$A\psi\le-1$ on $E\,;$ the equality 
$(dd^cu^\star_E)^n=0$ on $\Omega\setminus\ovl E$ in (c) is proved
exactly in the same way as 13.5~(a) in Step 1.
\medskip
\pritem{(d)} If $E\subset\{v=-\infty\}$, $v\in P(\Omega)$, $v\le 0$, then
for every $\varepsilon>0$ we have $\varepsilon v\le u_E$, hence $u_E=0$ on
$\Omega\setminus\{v=-\infty\}$ and $u^\star_E=0$.

Conversely, Choquet's lemma shows that there is an increasing sequence
$v_j\in P(\Omega)$, $-1\le v_j\le u_E$, converging almost everywhere to
$u^\star_E$. If $u^\star_E=0$, we can extract a subsequence in such a way that
$\int_\Omega |v_j|d\lambda<2^{-j}$.
As $v_j\le 0$ and $v_j\le-1$ on $E$, the function $v=\sum v_j$ is plurisubharmonic
$\le0$ and $v=-\infty$ on $E.$\qed}

\statement{(13.12) Proposition}{Let $\Omega\compact\bbbc^n$ and let
$K_1\supset K_2\supset\ldots$, $K=\bigcap K_j$ be compact subsets
of~$\Omega$. Then
\medskip
\item{\rm(a)} $\big(\buildu{j\to+\infty}\under\lim 
\uparrow u^\star_{K_j}\big)^\star=u^\star_K$.
\medskip
\item{\rm(b)} $\lim c(K_j,\Omega)=c(K,\Omega)$.
\medskip
\item{\rm(c)} $c^\star(K,\Omega)=c(K,\Omega)$.}

\proof{We have $\lim\uparrow u^\star_{K_j}\le u^\star_K$ by 13.11 (a).
On the other hand, let \hbox{$v\in P(\Omega)$} be such that $v\le 0$
on $\Omega$ and $v\le -1$ on $K$. For every $\varepsilon>0$ the open set
$\{v<-1+\varepsilon\}$ is a neighborhood of $K$, thus
$K\subset\{v<-1+\varepsilon\}$ for $j$ large. We obtain therefore
$v-\varepsilon\le u^\star_{K_j}$ and $u_K=\sup\{v\}\le\lim u^\star_{K_j}$,
whence equality (a). Property (b) follows now from Prop.~13.4 and 13.5~(b), 
and (c) is a consequence of (b) when the $K_j$'s are neighborhoods
of~$K$.\qed}

\corollary{13.13}{Every negligible set $N\subset\Omega$ satisfies
$c^\star(N,\Omega)=0$.}

\proof{By Choquet's lemma every negligible set is contained in a Borel
negligible set $N=\{v<v^\star\}$ with $v=\sup v_j$. In Step~2 of the
proof of Prop.~13.4, 13.5 and 13.6, we already showed that
$N\subset G\cup\bigcup K_{\delta\lambda\mu}$ with $G$ open,
$c(G,\Omega)<\varepsilon $ and $c(K_{\delta\lambda\mu},\Omega)=0$. By
definition $c^\star(G,\Omega)=c(G,\Omega)$, and by 13.12 (c) we also
have $c^\star(K_{\delta\lambda\mu},\Omega)=0$, therefore
$c^\star(N,\Omega)<\varepsilon $ for every $\varepsilon>0$.\qed}

We finally extend the basic formula 13.5~(b) to arbitrary relatively compact
subsets $E\compact\Omega$. We first need a lemma.

\lemma{13.14}{Let $\Omega\compact\bbbc^n$ and let $u,v\in P(\Omega)$ be
locally bounded plurisubharmonic functions such that $u\le v\le 0$ and 
$\buildu z\to\partial\Omega\under\lim u(z)=0$.
Then
$$\int_\Omega(dd^cv)^n\le\int_\Omega(dd^cu)^n.$$
Moreover $\int_\Omega(dd^cu)^n=0$ if and only if $u=0$.}

\proof{As $\max\{u+\varepsilon ,v\}=u+\varepsilon $ near $\partial\Omega
$, we get 
$$\int_\Omega(dd^cu)^n=\int_\Omega(dd^c\max\{u+\varepsilon ,v\})^n.$$
Let $\varepsilon $ tend to 0, and observe that the integrand on the
right hand side converges weakly to $(dd^cv)^n$ by Th.~1.7.
The asserted inequality follows.

Now, assume that $u(z_0)<0$ at some point. Then
$$v(z)=\max\{u(z),\varepsilon^2|z|^2-\varepsilon\}$$
coincides with $u$ near $\partial\Omega$ and with $\varepsilon
^2|z|^2-\varepsilon $ on a neighborhood $\omega$ of $z_0$. We get
therefore
$$\int_\Omega(dd^cu)^n=\int_\Omega(dd^cv)^n\ge
\int_\omega(dd^cv)^n>0.\eqno{\sq}$$}

\statement{(13.15) Proposition}{Let $\Omega\compact\bbbc^n$ be strongly
pseudoconvex. If $E\compact\Omega$ is an arbitrary subset, then
$$c^\star(E,\Omega)=\int_\Omega(dd^cu^\star_E)^n.$$}

\proof{We first show that
$$c(G,\Omega)=\int_{\ovl G}(dd^cu^\star_G)^n=\int_\Omega
(dd^cu^\star_G)^n\leqno(13.16)$$
for every open set $G\compact\Omega$. Let $K_1\subset K_2\subset\ldots$ be
compact subsets of $G$ with $K_j\subset K^0_{j+1}$ and $\bigcup K_j=G$.
Then $u^\star_{K_j}=-1$ on $K^0_j\supset K_{j-1}$ and $\lim\downarrow
u^\star_{K_j}=-1$ on $G$. Therefore $u^\star_G\le\lim u^\star_{K_j}\le
u_G\le u^\star_G$ and Th.~1.7, 13.5~(b), 13.11~(c) imply the above
equality~(13.16).

Now, let $E\compact\Omega$ be given and let $\psi<0$ be a strictly
plurisubharmonic exhaustion function on~$\Omega$. For every open set
$G\supset E$ with $G\compact\Omega$, we have $u^\star_G\ge A\psi$ and
$u^\star_E\ge u^\star_G$ by~13.11. Lemma~13.14 implies
$$\int_\Omega(dd^cu^\star_E)^n\le\int_\Omega(dd^cu^\star_G)^n=c(G,\Omega),$$
thus $\int_\Omega(dd^cu^\star_E)^n\le c^\star(E,\Omega)$.

Conversely, Choquet's lemma shows that there exists an increasing
sequence $v_j\in P(\Omega)$ with $-1\le v_j\le0$, $v_j\ge A\psi $ on
$\Omega$ and $\lim v_j=u_E$ almost everywhere. If
$$G_j=\{z\in\Omega ;(1+1/j)v_j(z)<-1\}$$
then $G_j\supset E$, $G_j$ is decreasing and $(1+1/j)v_j\le u_{G_j}$.
Thus $\lim\uparrow u^\star_{G_j}=u^\star_E$ almost everywhere and Th.~13.4
gives 
$$\lim_{j\to+\infty}\int_\Omega(dd^cu^\star_{G_j})^n=\int_\Omega
(dd^cu^\star_E)^n.\eqno{\sq}$$}

\corollary{13.17}{Let $\Omega\compact\bbbc^n$ be strongly
pseudoconvex. If $E\compact\Omega $, then $c^\star(E,\Omega)=0$ if and only
if $u^\star_E=0.$\qed}

\titleb{14. Outer Regularity of Monge-Amp\`ere Capacities and\hfill\break
\phantom{14.\ }Characterization of Pluripolar and Negligible Sets}
\titlebrunning{Outer Regularity and Pluripolar Sets}
We first check that the outer Monge-Amp\`ere capacity
$c^\star$ is actually a capacity.

\theorem{14.1}{Let $\Omega\compact\bbbc^n$ be strongly pseudoconvex. 
Then the outer capacity $c^\star(\bu,\Omega)$ is a capacity
in the sense of Def. $10.1$.}

\proof{Axiom 10.1 (a) is clear, and 10.1 (c) is a consequence of 13.12.
To prove 10.1~(b), we only have to show that $c^\star(\bigcup E_j,\Omega
)\le\lim_{j\to+\infty}c^\star(E_j,\Omega)$.
It is no loss of generality to assume that $E_j\compact\Omega$. Let
$N_j$ be the negligible set $N_j=\{u_{E_j}<u^\star_{E_j}\}$ and $G_0$ an
open subset of $\Omega$ with $G_0\supset\bigcup N_j$ and $c(G_0,\Omega)
<\varepsilon$ (such an open set exists by Cor.~13.13). Consider the open
sets defined by $V_j=\{u^\star_{E_j}<-1+\eta\}$ and $G_j=G_0\cup V_j\supset
E_j$. Then $(1-\eta)^{-1}u^\star_{E_j}\le u^\star_{V_j}\le 0$ and
Lemma~13.14 implies
$$\eqalign{
c(G_j,\Omega) &\le\varepsilon+c(V_j,\Omega)
=\varepsilon+\int_\Omega(dd^cu^\star_{V_j})^n\cr
&\le\varepsilon+(1-\eta)^{-n}\int_\Omega(dd^cu^\star_{E_j})^n=
\varepsilon+(1-\eta)^{-n}c^\star(E_j,\Omega),\cr}$$
thanks to Prop.~13.15. Further $E_j\subset G_j$ and $G_1\subset
G_2\subset\ldots $ since $u^\star_{E_j}$ is decreasing. Thus $G=\bigcup
G_j\supset E=\bigcup E_j$ and
$$c(G,\Omega)=\lim_{j\to+\infty}c(G_j,\Omega)\le
\varepsilon+(1-\eta)^{-n}\lim_{j\to+\infty}c^\star(E_j,\Omega).$$
Letting $\varepsilon ,\eta\to 0$ we get the desired inequality
$$c^\star(E,\Omega)\le\lim_{j\to+\infty}c^\star(E_j,\Omega).\eqno{\sq}$$}

\corollary{14.2}{If $\Omega\compact\bbbc^n$ is strongly pseudoconvex, then
$c(\bu,\Omega)$ is outer regular and every $K$-analytic subset
$($in particular every Borel subset$)$ of $\Omega$ is capacitable.}

\proof{Choquet's capacitability theorem combined with 13.12~(c) implies that
every $K$-analytic subset $E\subset\Omega$ satisfies 
$$c^\star(E,\Omega)=\sup_{K\,{\rm compact}\,\subset E}c^\star(K,\Omega)=
\sup_{K\,{\rm compact}\,\subset E}c(K,\Omega)=c_\star(E,\Omega).\eqno\sq$$}

Now, we prove an important result due to Josefson [Jo]. A set $P$ in
$\bbbc^n$ is said to be {\it locally pluripolar} if for each $z\in P$
there is an open neighborhood $\Omega$ of $z$ and $v\in P(\Omega)$
such that $P\cap\Omega\subset\{v=-\infty\}$.

\statement{(14.3) Theorem {\rm(Josefson)}}{If $P\subset\bbbc^n$ is locally
pluripolar, there exists\break $v\in P(\bbbc^n)$ with
$P\subset\{v=-\infty\}$, i.e.\ $P$ is globally pluripolar in $\bbbc^n$.
Moreover, $v$ can be taken to have logarithmic growth at infinity,
such that $v(z)\le\log_+|z|$.}

\proof{By the definition of locally pluripolar, we can find sets
$P_j,\Omega_j$ with $\Omega_j$ open and strictly pseudoconvex,
$P_j\compact\Omega_j\compact\bbbc^n$, $\bigcup_{j\ge1}P_j=P$ and
such that $P_j$ is contained in the $-\infty$ poles of a single
plurisubharmonic function in $\Omega_j$. By 13.11~(d)
and 13.15, we have $c^\star(P_j,\Omega_j)=0$.

Let $B_k$ be the ball of center 0 and radius $e^{2^k}$ in $\bbbc^n$ and
let $j(k)$ be a sequence of integers such that each integer is repeated
infinitely many times and $\Omega_{j(k)}\subset B_k$.
By the comparison result (11.3) we have
\hbox{$c^\star(P_{j(k)},B_{k+1})=0$,} hence the extremal function
$u^\star_{P_{j(k)}}$ in $B_{k+1}$ is zero and we can find
\hbox{$v_k\in P(B_{k+1})$} with $-1\le v_k\le 0$, $v_k=-1$ on $P_{j(k)}$
and $\int_{B_k}|v_k|d\lambda<2^{-k}$. Now set
$$\wt v_k(z)=\cases{
v_k(z) &\quad on $\;\;B_k$\cr
\max\{v_k(z),2^{-k}\log|z|-2\} &\quad on $\;\;B_{k+1}\setminus B_k$\cr
2^{-k}\log|z|-2&\quad on $\;\;\bbbc^n\setminus B_{k+1}.$\cr}$$
As $\wt v_k\le0$ on $B_k$ and $\int_{B_k}|\wt v_k|d\lambda
<2^{-k}$,
the series $v=\sum\wt v_k$ converges to a global plurisubharmonic
function $v\not\equiv-\infty$ on $\bbbc^n$.
Moreover $\wt v_k=-1$ on $P_{j(k)}$ and each $P_j$ is repeated
infinitely many times, therefore
$$v=-\infty\quad{\rm on}\;\;\bigcup P_j=P.$$
Finally, we have $v(z)\le\sum 2^{-k}\log_+|z|\le\log_+|z|$ (the logarithmic
growth estimate was first obtained by H.~El~Mir [E-M1] and J.~Siciak [Sic]).}

\corollary{14.4}{Let $\Omega\subset\bbbc^n$ and $P\subset\Omega$.
Then $P$ is pluripolar in $\Omega$ if and only if $c^\star(P,\Omega)=0$.}

\proof{That $P$ is pluripolar implies $c^\star(P,\Omega)=0$ was proved in
Cor.~11.6. Conversely, if $c^\star(P,\Omega)=0$ then
$c^\star(P\cap\omega',\omega)=0$ for all concentric balls
$\omega '\compact\omega\compact\Omega$ and Cor.~13.17 combined with
13.11 (d) shows that $P\cap\omega'$ is pluripolar in $\omega$. Josefson's
theorem implies that $P$ is globally pluripolar in~$\bbbc^n.$\qed}

\corollary{14.5}{Negligible sets are the same as pluripolar sets.}

\proof{We have already seen that pluripolar sets are negligible (by
Prop.\ 13.2 applied in $\bbbc^n$). Conversely, a negligible set $N\subset
\Omega$ satisfies $c^\star(N,\Omega)=0$ by Cor.~13.13, hence $N$ is
pluripolar.\qed}

\titleb{15. Siciak Extremal Functions and Alexander Capacity}
We work here on the whole space $\bbbc^n$ rather than on a bounded open
subset~$\Omega$. In this case, the relevant class of plurisubharmonic
functions to consider is the set $P_{\rm log}(\bbbc^n)$ of
plurisubharmonic functions $v$ with logarithmic growth at infinity,
i.e.\ such that 
$$v(z)\le\log_+|z|+C\leqno(15.1)$$
for some real constant $C$. Let $E$ be a bounded subset of $\bbbc^n$.
We consider the global extremal function introduced by Siciak [Sic]:
$$U_E(z)=\sup\{v(z)\,;~v\in P_{\log}(\bbbc^n),\; v\le0~\hbox{\rm on}~
E\}.\leqno(15.2)$$

\theorem{15.3}{Let $E$ be a bounded subset of~$\bbbc^n$. We have
$U_E^\star\equiv+\infty$ if and only if $E$ is pluripolar. Otherwise,
$U^\star_E\in P_{\log}(\bbbc^n)$ and $U^\star_E$ satisfies an inequality
$$\log_+(|z|/R)\le U^\star_E(z)\le\log_+|z|+M$$
for suitable constants $M,R>0$. Moreover $U^\star_E=0$ on $E^0$,
$$(dd^cU^\star_E)=0~{\rm on}~\bbbc^n\setminus\ovl E,\qquad
\int_{\ovl E}(dd^cU^\star_E)^n=1.$$}

\proof{If $E$ is pluripolar, Th.~14.3 shows that
there exists $v\in P_{\log}(\bbbc^n)$ such that $v=-\infty$ on $E$. Then
$v+C\le 0$ on $E$ for every $C>0$ and we get $U_E=+\infty$ on
$\bbbc^n\ssm v^{-1}(-\infty)$, thus $U_E^\star\equiv+\infty$. Conversely,
if $U_E^\star\equiv+\infty$, there is a point $z_0$ in the unit ball
$B\subset\bbbc^n$ such that $U_E(z_0)=+\infty$. Therefore, there is a
sequence of functions $v_k\in P_{\log}(\bbbc^n)$ with
$v_k\le 0$ on $E$ and $M_k=\sup_B v_k\to +\infty$. By taking a
suitable subsequence we may assume $M_k\ge 2^{k+1}$. We claim that
$w=\sum_{k\in\bbbn}2^{-k-1}(v_k-M_k)$ satisfies $w=-\infty$ on~$E$
(obvious), $w\in P_{\log}(\bbbc^n)$ and $w\not\equiv-\infty\,$; this will
imply that $E$ is actually pluripolar. In fact, if $v$ is plurisubharmonic
on $\bbbc^n$, we have $\sup_{B(0,r)}v=\chi(\log r)$ where $\chi$ is a
convex increasing function, and the condition $v\in P_{\log}(\bbbc^n)$
implies $\chi'\le 1$ on $\bbbr\,$; therefore
$$\chi(\log r)\le\chi(\log r_0)+\log r/r_0\quad\hbox{\rm for}~r\ge r_0.$$
In our case, by taking $v=v_k-M_k$ and $r_0=1$, we find
$v_k(z)-M_k\le\log_+|z|$, thus $w(z)\le\log_+|z|$ and $w\in P_{\log}(\bbbc^n)$.
Moreover, by the maximum principle, there is a point $z_k\in\partial B$ at
which $v_k(z_k)-M_k=0$. The Harnack inequality for subharmonic functions
applied to the nonpositive function $v=v_k-M_k-\log 2$ on $B(0,2)\subset
\bbbr^{2n}$ shows that
$$\eqalign{0
&\ge\int_{S(0,2)}(v_k(z)-M_k-\log 2)\,d\sigma(z)\cr
&=\int_{S(0,2)}v(z)\,d\sigma(z)\ge -C\,v(z_k)=-C\log 2\cr}$$
where $C=(1+1/2)^{2n}/(1-1/4)>0$ and $d\sigma$ is the unit invariant
measure on the sphere. Hence $\int_{S(0,2)}w(z)\,d\sigma(z)>-\infty$,
$w\not\equiv-\infty$, and $E$ is pluripolar.

Now, assume that $E$ is not pluripolar. The above arguments show that there
must exist a uniform upper bound $\sup_B v\le M$ for all functions
$v\in P_{\log}(\bbbc^n)$ with $v\le 0$ on~$E$, thus $v(z)\le\log_+|z|+M$
and $U_E^\star(z)=(\sup\{v\})^\star\le\log_+|z|+M$. This implies that
$U_E^\star\in P_{\log}(\bbbc^n)$. On the other hand, $E$ is contained in a
ball $\ovl B(0,R)$ so $\log_+|z|/R\le 0$ on $E$ and we get
$U_E(z)\ge\log_+|z|/R$. The equality $(dd^cU^\star_E)^n=0$ on
$\bbbc^n\setminus\ovl E$ is verified exactly in the same way as 12.8 (a).
The value of the integral of $(dd^cU^\star_E)^n$ over $\ovl E$ is obtained
by the following lemma.\qed}

\lemma{15.4}{Let $v\in P_{\log}(\bbbc^n)$ be such that
$$\log_+|z|-C_1\le v(z)\le\log_+|z|+C_2$$
for some constants. Then $\int_{\bbbc^n}(dd^cv)^n=1$.}

\proof{It is sufficient to check that
$$\int_{\bbbc^n}(dd^cv_1)^n\le\int_{\bbbc^n}(dd^cv_2)^n$$
when $v_1,v_2$ are two such functions. Indeed, we have
$$\int_{\bbbc^n}(dd^c\log_+|z|)^n=
\int_{\bbbc^n}(dd^c\log|z|)^n=1$$
by Stokes' theorem and remark 3.10, and we only have to choose $v_1(z)$
or $v_2(z)=\log_+|z|$ and the other function equal to $v$. To prove the
inequality, fix $r,\varepsilon>0$ and choose $C>0$ large enough so that
$(1-\varepsilon)v_1>v_2-C$ on $B(0,r)$.
As the function
$u=\max\{(1-\varepsilon)v_1,v_2-C\}$ is equal to $v_2-C$ for $|z|=R$
large, we get 
$$(1-\varepsilon)^n\int_{B(0,r)}\!(dd^cv_1)^n=
\int_{B(0,r)}\!(dd^cu)^n\le
\int_{B(0,R)}\!(dd^cu)^n=\int_{B(0,R)}\!(dd^cv_2)^n$$
and the expected inequality follows as
$\varepsilon\to0$ and $r\to+\infty.$\qed}

\theorem{15.5}{Let $E,E_1,E_2,\ldots\subset B(0,R)\subset\bbbc^n$.
\medskip
\item{\rm(a)} If $E_1\subset E_2$, then $U^\star_{E_1}\ge U^\star_{E_2}$.
\medskip
\item{\rm(b)} If $E_1\subset E_2\subset\ldots$ and $E=\bigcup E_j$, then
$U^\star_E=\lim\downarrow U^\star_{E_j}$.
\medskip
\item{\rm(c)} If $K_1\supset K_2\supset\ldots$ and $K=\bigcap K_j$, then
$U^\star_K=(\lim\uparrow U^\star_{K_j})^\star$.
\medskip
\item{\rm(d)} For every set $E$, there exists a decreasing
sequence of open sets $G_j\supset E$ such that $U^\star_E=(\lim\uparrow
U^\star_{G_j})^\star$.
\vskip0pt}

\proof{(a) is obvious and the proof of (c) is similar to that of 13.12 (a).
\medskip
\pritem{(d)} By Choquet's lemma, there is an increasing sequence $v_j\in
P_{\log}(\bbbc^n)$ with $U^\star_E=(\lim v_j)^\star$ and
$v_j(z)\ge\log_+|z|/R$.
Set $G_j=\{v_j<1/j\}$ and observe that $U^\star_{G_j}\ge v_j-1/j$.
\medskip
\pritem{(b)} Set $v=\lim\downarrow U^\star_{E_j}$. Then $v\in P_{\log}(\bbbc^n)$
and $v=0$ on $E$, except on the negligible set
$N=\bigcup\{U_{E_j}<U^\star_{E_j}\}$.
By Josefson's theorem~14.3, there exists $w\in P(\bbbc^n)$ such that 
$N\subset\{w=-\infty\}$ and $w(z)\le\log_+(|z|/R)$, in particular
$w\le 0$ on~$E$. We set
$$v_j(z)=\Big(1-{1\over j}\Big)v(z)+{1\over j}w(z).$$
Then $v_j\in P_{\log}(\bbbc^n)$ and $v_j\le 0$ everywhere on~$E$. Therefore
$$U^\star_E\ge U_E\ge v_j=\Big(1-{1\over j}\Big)v+{1\over j}w
~~{\rm on}~\cb^n$$
and letting $j\to+\infty$ we get $U^\star_E\ge v$. The converse inequality
$U^\star_E\le\lim\downarrow U^\star_{E_j}$ is clear.\qed}

Now, we show that the extremal function of a compact set can be computed
in terms of polynomials. We denote by ${\cal P}_d$ the space of
polynomials of degree $\le d$ in $\bbbc[z_1\ld z_n]$.

\theorem{15.6}{Let $K$ be a compact subset of $\bbbc^n$. Then
$$U_K(z)=\sup\Big\{
{1\over d}\log|P(z)|\,;\,d\ge 1,\,
P\in{\cal P}_d,\,\|P\|_{L^\infty(K)}\le1\Big\}.$$}

\proof{For any of the polynomials $P$ involved in the above formula, we
clearly have ${1\over d}\log|P|\in P_{\log}(\bbbc^n)$ and this function is
$\le 0$ on $K$. Hence
$${1\over d}\log|P|\le U_K.$$
Conversely, fix a point $z_0\in\bbbc^n$ and a real number $a<U_K(z_0)$. Then
there exists $v\in P_{\log}(\bbbc^n)$ such that $v\le 0$ on $K$ and
$v(z_0)>a$. Replacing $v$ by $v\star\rho_\delta -\varepsilon $ with $\delta
\ll\varepsilon\ll 1$, we may assume that $v\in
P_{\log}(\bbbc^n)\cap\ci(\bbbc^n)$, $v<0$ on $K$ and $v(z_0)>a$. Choose a ball
$B(z_0,r)$ on which $v>a$, a smooth function $\chi $ with compact support
in $B(z_0,r)$ such that $\chi=1$ on $B(z_0,r/2)$ and apply H\"ormander's
$L^2$ estimates to the closed $(0,1)$-form $d''\chi$ 
and to the weight
$$\varphi(z)=2dv(z)+2n\log|z-z_0|+\varepsilon\log(1+|z|^2).$$
We find a solution $f$ of $d''f=d''\chi$ such that
$$\eqalign{
\int_{\bbbc^n}|f|^2e^{-2dv}&|z-z_0|^{-2n}(1+|z|^2)^{-\varepsilon}d\lambda
\cr
&\le\int_{B(z_0,r)}|d''\chi|^2e^{-2dv}|z-z_0|^{-2n}
(1+|z|^2)^{2-\varepsilon}d\lambda\le C_1e^{-2da}.\cr}$$
We thus have $f(z_0)=0$ and $F=\chi-f$ is a holomorphic function on
$\bbbc^n$ such that $F(z_0)=1$. In addition we get
$$\int_{\bbbc^n}|F|^2e^{-2dv}(1+|z|^2)^{-2n-2\varepsilon}d\lambda\le
C_2e^{-2da}$$
where $C_1,C_2>0$ are constants independent of $d$. As
$v(z)\le\log_+|z|+C_3$, it follows that $F\in{\cal P}_d$.
Moreover, since $v<0$ on a neighborhood of $K$, the mean value
inequality applied to the subharmonic function $|F|^2$ gives
$$\sup_K|F|^2\le C_4e^{-2da}.$$
The polynomial $P=C^{-1/2}_4e^{da}F\in{\cal P}_d$ is such that
$\|P\|_K=1$ and we have $\log|P(z_0)|\ge da-C_5$, whence 
$$\sup\Big\{{1\over d}\log|P(z_0)|\,;\,d\ge 1,\,
P\in{\cal P}_d,\,\|P\|_{L^\infty(K)}\le1\Big\}\ge a.$$
As $a$ was an arbitrary number $< U_K(z_0)$, the proof is complete.\qed}

Now, we introduce a few concepts related to extremal polynomials. Let
$B$ be the unit ball of $\bbbc^n$ and $K$ a compact subset of $B$.
The Chebishev constants $M_d(K)$ are defined by 
$$M_d(K)=\inf\big\{\|P\|_{L^\infty(K)}\; ;\;
P\in{\cal P}_d,\,\|P\|_{L^\infty(B)}=1\big\}.\leqno(15.7)$$
It is clear that $M_d(K)\le 1$ and that $M_d(K)$ satisfies
$$M_{d+d'}(K)\le M_d(K)M_{d'}(K).$$
The {\it Alexander capacity} is defined by
$$T(K)=\inf_{d\ge 1}M_d(K)^{1/d}.\leqno(15.8)$$
It is easy to see that we have in fact
$T(K)=\lim_{d\to+\infty}M_d(K)^{1/d}:$
for any integer $\delta\ge1$, write $\delta=qd+r$ with $0\le r<d$ and
observe that 
$$M_\delta(K)^{1/\delta}\le M_{qd}(K)^{1/(qd+r)}\le
M_d(K)^{q/(qd+r)}\,;$$
letting $\delta\to+\infty$ with $d$ fixed, we get
$$T(K)\le\liminf_{\delta\to+\infty}M_\delta(K)^{1/\delta}\le
\limsup_{\delta\to+\infty}M_\delta(K)^{1/\delta}\le
M_d(K)^{1/d},$$
whence the equality. Now, for an arbitrary subset $E\subset B$, we set
$$T_\star(E)=\sup_{K\subset E}T(K)\, ,\qquad
T^\star(E)=\inf_{G\,{\rm open}\,\supset E}T_\star(G).\leqno(15.9)$$

\vbox{\statement{(15.10) Siciak's theorem}{For every set $E\subset B$,
$$T^\star(E)=\exp(-\sup_BU^\star_E).$$}}

\proof{The main step is to show that the equality holds for compact
subsets $K\subset B$, i.e.\ that
$$T(K)=\exp(-\sup_BU^\star_K).\leqno(15.11)$$
Indeed, it is clear that $\sup_BU^\star_K=\sup_BU_K$ and Th.~15.6 gives
$$\eqalign{
\sup_BU_K &=\sup\big\{
{1\over d}\log\|P\|_{L^\infty(B)}\; ;\;
d\ge1,\, P\in{\cal P}_d,\,\|P\|_{L^\infty(K)}=1\big\}\cr
&=\sup\big\{
-{1\over d}\log\|P\|_{L^\infty(K)}\; ;\;
d\ge1,\, P\in{\cal P}_d,\,\|P\|_{L^\infty(B)}=1\big\}\cr}$$
after an obvious rescaling argument $P\mapsto\alpha P$. Taking the
exponential, we get
$$\eqalign{
\exp(-\sup_BU^\star_K) &=
\inf_{d\ge1}\inf\big\{\|P\|^{1/d}_{L^\infty(K)}\; ;\;
P\in{\cal P}_d,\,\|P\|_{L^\infty(B)}=1\big\}\cr
&=\inf_{d\ge1}M_d(K)^{1/d}=T(K).\cr}$$

Next, let $G$ be an open subset of $B$ and $K_j$ an increasing sequence
of compact sets such that $G=\bigcup K_j$ and
$T_\star(G)=\lim T(K_j)$. Then 15.5 (b) implies $U^\star_G=\lim\downarrow
U^\star_{K_j}$, hence
$$\lim_{j\to+\infty}\sup_B U^\star_{K_j}=
\sup_B U^\star_G=\sup_{\ovl B}U^\star_G$$
by Dini's lemma. Taking the limit in (15.11), we get 
$$T_\star(G)=\exp(-\sup_BU^\star_G).$$
Finally, 15.5 (d) shows that there exists a decreasing sequence of open
sets $G_j\supset E$ such that $U^\star_E=(\lim\uparrow U^\star_{G_j})^\star$.
We may take $G_j$ so small that $T^\star(E)=\lim T_\star(G_j)$.
Theorem 15.9 follows.\qed}

\corollary{15.12}{The set function $T^\star$ is a capacity in
the sense of {\rm Def.~10.1} and we have
$T^\star(E)=T_\star(E)$ for every $K$-analytic set $E\subset B$.}

\proof{Axioms 10.1 (a,b,c) are immediate consequences of properties 15.5
(a,b,c) respectively. In addition, formulas 15.10 and 15.11 show that we
have $T^\star(K)=T(K)$ for every compact set $K\subset B$. The last 
statement is then a consequence of Choquet's capacitability theorem.\qed}

To conclude this section, we show that $1/|\log T^\star|$ is not very far
from being subadditive. We need a lemma.

\lemma{15.13}{For every $P\in{\cal P}_d$, one has
$$\log\|P\|_{L^\infty(B)}-c_nd\le
\int_{\partial B}\log|P(z)|d\sigma(z)\le\log\|P\|_{L^\infty(B)}$$
where $d\sigma$ is the unit invariant measure on the sphere and $c_n$ a
constant such that $c_n\sim\log(2n)$ as $n\to+\infty$.}

\proof{Without loss of generality, we may assume that
$\|P\|_{L^\infty(B)}=1$. Since
${1\over d}\log|P|\in P_{\log}(\bbbc^n)$,
the logarithmic convexity property already used implies that
$$\sup_{B(0,r)}{1\over d}\log|P|\ge\log r\quad{\rm for}~r<1.$$
The Harnack inequality for the Poisson kernel now implies
$$\sup_{B(0,r)}\log|P|\le
{1-r^2\over(1+r)^{2n}}\int_{\partial B}\log|P|d\sigma ,$$
$$\int_{\partial B}\log|P|d\sigma\ge
{1+r^{2n}\over 1-r^2}\log r.d.$$
The lemma follows with
$$c_n=\inf_{r\in{}]0,1[}
{(1+r)^{2n}\over 1-r^2}\log{1\over r}\,;$$
the infimum is attained approximately for 
$r=1/(2n\log 2n).$\qed}

\corollary{15.14}{For
$P_j\in{\cal P}_{d_j},\; 1\le j\le N$,
$$\|P_1\ldots P_N\|_{L^\infty(B)}\ge
e^{-c_n(d_1+\cdots+d_N)}
\|P_1\|_{L^\infty(B)}\ldots\|P_N\|_{L^\infty(B)}.$$}

\proof{Apply Lemma~15.13 to each $P_j$ and observe that
$$\int_{\partial B}\log|P_1\ldots P_N|d\sigma=
\sum_{1\le j\le N}\int_{\partial B}\log|P_j|d\sigma.\eqno{\sq}$$}

\theorem{15.15}{For any set $E=\bigcup_{j\ge1}E_j$, one has
$${1\over c_n-\log T^\star(E)}\le\sum_{j\ge1}
{1\over |\log T^\star(E_j)|}.$$}

\proof{It is sufficient to check the inequality for a finite union
$K=\bigcup K_j$ of compacts sets $K_j\subset B$, $1\le j\le N$. Select
$P_j\in{\cal P}_{d_j}$ such that
$$\|P_j\|_{L^\infty(B)}=1,\qquad
\|P_j\|_{L^\infty(K_j)}=M_{d_j}(K_j),$$
and set $P=P_1\ldots P_N,\; d=d_1+\cdots+d_N$.
Then Cor.~15.14 shows that $\|P_j\|_{L^\infty(B)}\ge e^{-c_nd}$,
thus
$$M_d(K)\le e^{c_nd}\|P\|_{L^\infty(K)}.$$
If $z\in K$ is in $K_j$, then
$|P(z)|\le|P_j(z)|\le\|P_j\|_{L^\infty(K_j)}$
because all other factors are $\le 1$. Thus
$$M_d(K)\le e^{c_nd}\max\{\|P_j\|_{L^\infty(K_j)}\},$$
$$T(K)\le M_d(K)^{1/d}\le e^{c_n}\max\{M_{d_j}(K)^{1/d_j\cdot
d_j/d}\}.$$
Take $d_j=[k\alpha_j]$ with arbitrary $\alpha_j>0$ and let
$k\to+\infty$.
It follows that
$$T(K)\le e^{c_n}\max\{T(K_j)^{\alpha_j/\alpha}\}$$
where $\alpha=\sum\alpha_j$. The inequality asserted in Th.~15.15 
is obtained for the special choice
$\alpha_j=1/|\log T(K_j)|>0$ which makes all terms in
$\max\{\ldots\}$ equal.\qed}

\titleb{16. Comparison of Capacities and El Mir's Theorem}
We first prove a comparison theorem for the capacities $c(\bu,\Omega)$
and $T$, due to Alexander and Taylor [A-T].

\theorem{16.1}{Let $K$ be a compact subset of the unit ball
$B\subset\bbbc^n$. Then
\medskip
\item{\rm(a)} $T(K)\le\exp(-c(K,B)^{-1/n})$.
\medskip
\pritem{}For each $r<1$, there is a constant $A(r)$ such that
\medskip
\item{\rm(b)} $T(K)\ge\exp(-A(r)\,c(K,B)^{-1})$
when $K\subset B(0,r)$.}

\remark{16.2}{Both set functions $c^\star(\bu,\Omega)$ and $T^\star$
are capacities in the sense of~10.1. Hence, the estimates of
the theorem also hold for all $K$-analytic sets, in particular all
Borel sets.}

\remark{16.3}{The inequalities are sharp, at least as far as the
exponents on $c(K,B)$ are concerned. For if $K=\ovl B(0,\varepsilon)$,
then it is easy to check that $T(K)=\varepsilon $ and example 13.10
gives $c(K,B)=(\log\,1/\varepsilon)^{-n}$. Hence, equality holds in
16.1~(a). On the other hand, if $K$ is a small polydisc
$$K=\{(z_1\ld z_n)\in\bbbc^n\; ;\;
|z_1|\le\delta ,\,
|z_j|\le 1/n,\,
1<j\le n\}$$
and $\delta\le 1/n$, then $T(K)\le\delta $, while $c(K,B)\ge
C(\log\,1/\delta)^{-1}$.
To check this last inequality (which shows the optimality of 16.1~(b)), put
$$u(z)=\Big(\log{1\over\delta}\Big)^{-1}\log_+{|z_1|\over\delta}+
(\log n)^{-1}\sum^n_{j=2}\log_+(n|z_j|).$$
Then $u=0$ and $u\le n$ on $B$, hence
$$c(K,B)\ge\int_B\Big({1\over n}dd^cu\Big)^n=
{n!\over n^n}(\log n)^{-(n-1)}(\log\,1/\delta)^{-1}$$
because all measures $dd^c\log_+(|z_j|/r)$ have total mass 1 in $\bbbc$.}

\pproof{Proof of Theorem 16.1}{If $K$ is pluripolar then $U_K^\star\equiv
+\infty$, hence $c(K,B)=T(K)=0$ and the inequalities are satisfied. We thus
assume that $K$ is not pluripolar.}

\pproof{Proof of 16.1 (a)}{Set $M=\sup_BU^\star_K\,;$ then $T(K)=e^{-M}$ by
Siciak's theorem. Since $u=U^\star_K/M\in P(B)$ and $0\le u\le1$ on $B$, we
get
$$c(K,B)\ge M^{-n}\int_K(dd^cU^\star_K)^n=M^{-n}$$
by Th.~15.3. This inequality is equivalent to 16.1 (a).\qed}

\pproof{Proof of 16.1 (b)}{Let $u^\star_K$ be the extremal function for $K$
relative to the ball $B'=B(0,e)\supset\!\supset B$. For any
$v\in P_{\log}(\bbbc^n)$ such that $v\le 0$ on $K$, we have
$v\le U^\star_K\le M+1$ on $B'$, hence the function
$$w={v-M-1\over M+1}$$
satisfies $w\le 0$ on $B'$ and $w\le -1$ on $K$. We infer
$w\le u^\star_K\,;$ by taking the supremum over all choices of $v$, we get
$$u^\star_K\ge{U^\star_K-M-1\over M+1}.$$
Now, there is a point $z_0\in\ovl B$ such that $U^\star_K(z_0)=M$, thus
$$u^\star_K(z_0)\ge -{1\over M+1}.$$
As $u^\star_K\le0$ on $B'$, the mean value and Harnack inequalities show that
$$u^\star_K(z_0)\le C_1\int_{B'}u^\star_Kd\lambda\Longrightarrow
\|u^\star_K\|_{L^1(B')}\le-{1\over C_1}u^\star_K(z_0)\le
{C_2\over M}.$$
The Chern-Levine-Nirenberg inequalities 1.3 and 1.4 (a) imply now
$$c(K,B')=\int_B(dd^cu^\star_K)^n\le
C_3\|u^\star_K\|_{L^1(B')}\,\|u^\star_K\|^{n-1}_{L^\infty(B')}
\le{C_4\over M}.$$
As $K\subset B(0,r)\compact B$, Prop.~11.3 (b) gives
$$c(K,B)\le C_5(r)c(K,B')\le A(r)M^{-1}$$
and inequality 16.1~(b) follows.\qed}
             
We now prove El Mir's theorem [E-M1]. This result is an effective
version of Josefson's theorem: given a plurisubharmonic function in the ball,
a subextension can be found with prescribed singularities of poles
and slow growth at infinity.

\statement{(16.4) El Mir's theorem}{Let $v\in P(B)$ with $v\le -1$, 
$\varepsilon\in~]0,1/n[$ and $r<1$. Then there exists $u\in
P_{\log}(\bbbc^n)$ such that $u\le -|v|^{{1\over n}-\varepsilon}$ on
$B(0,r)$.}

\proof{For $t\ge1$, set $G_t=\{z\in B(0,r);v(z)<-t\}$ and let $U^\star_t\in
P_{\log}(\bbbc^n)$ be the Siciak extremal function of $G_t$. Since $G_t$ is
open, we have $U^\star_t=0$ on $G_t$. We set $M(t)=\sup_BU^\star_t$ and
$$u(z)=\varepsilon^{-1}\int^{+\infty}_1t^{-1-\varepsilon}
(U^\star_t(z)-M(t))dt.$$
Proposition~11.4 shows that $c(G_t,B)\le C_1/t$, therefore
$$M(t)=-\log T^\star(G_t)\ge c(G_t,B)^{-1/n}\ge C_2t^{1/n}$$
by inequality 16.1~(a). As $U^\star_t-M(t)\le0$ on $B$, we get
$U^\star_t(z)-M(t)\le\log_+|z|$ by logarithmic convexity, thus
$$u(z)\le\log_+|z|.$$
For $z\in B(0,r)$ we have $U^\star_t(z)=0$ as soon as $G_t\ni z$, i.e.\
$t<-v(z)$. Hence
$$u(z)\le-\varepsilon^{-1}\!\int^{|v(z)|}_1\!t^{-1-\varepsilon}M(t)dt\le
-C_3\int^{|v(z)|}_1\!t^{-1-\varepsilon+{1\over n}}dt=
-C_4|v(z)|^{{1\over n}-\varepsilon}.$$
Starting if necessary with a smaller value of $\varepsilon $ and
subtracting a constant to~$u$, we can actually get
$$u\le-|v|^{{1\over n}-\varepsilon}~~{\rm on}~B(0,r).$$
It remains to check that $u$ is not identically $-\infty$. By
logarithmic convexity again, we have
$$\sup_{\ovl B(0,1/2)}U^\star_t\ge M(t)-\log 2$$
and there exists $z_0\in S(0;1/2)$ such that 
$U^\star_t(z_0)-M(t)\ge-\log 2$. The Harnack inequality shows that
$${1-1/4\over(1+1/2)^{2n}}\int_{\partial B}
\big(U^\star_t(z)-M(t)\big)d\sigma(z)\ge
U^\star_t(z_0)-M(t)\ge -\log 2$$
and integration with respect to $t$ yields
$$\int_{\partial B}u(z)d\sigma(z)\ge -4/3(3/2)^{2n}\log 2>-\infty.
\eqno{\sq}$$}
\vskip20pt

\centerline{\tbfontt Complementary references}
\vskip12pt
\eightpoint
\lettre C-L-N|

\article A-T|H. Alexander {\rm and} B.A. Taylor|Comparison of two
     capacities in $\bbbc^n$|Math. Zeitschrift|186|1984|407--417|
\livre Ceg|U. Cegrell|Capacities in complex analysis|Aspects of
     Mathematics, Vol.~14, Vieweg \& Sohn, 153~p|1988|
\article Duf|A. Dufresnoy|Sur l'\'equation de Monge-Amp\`ere complexe
     dans la boule de $\bbbc^n$|Ann. Inst. Fourier|39|1989|773-775|     
\article E-M1|H. El Mir|Fonctions plurisousharmoniques et ensembles 
     polaires|S\'emi\-naire P.~Lelong~-- H.~Skoda (Analyse) 
     {\oldstyle1978{\rm/}79}, Lecture Notes in Math. no~822|\rm
     Springer-Verlag|1979|221--237|
\article Jo|B. Josefson|On the equivalence between locally polar and
     globally polar for plurisubharmonic functions on $\bbbc^n$|Ark. for
     Mat.|16|1978|109--115|
\livre Rud|W. Rudin|Function theory in the unit ball of
     $\bbbc^n$|Grundlehren der math. Wissenschaften, 241,
     Springer-Verlag|1980|
\livre Sic|J. Siciak|Extremal plurisubharmonic functions and capacities in
     $\bbbc^n$|Sophia Kokyuroku in Mathematics|1982|
\article Za|V. Zaharjuta|Transfinite diameter, 
     \v Ceby\v sev constants and capacity for compacta
     in $\bbbc^n$|Math. USSR-Sbornik|25|1975|350--364|
\vskip5pt
\tenpoint
\noindent(October 2, 1991, revised on May 9, 1995)

\end

