Theorem 5.47.10 (Josefson)1822741C4B07

Proof. Step 1: the glued functions. Choose compact sets $L_1,L_2,\ldots $ whose interiors cover $\mathbb C^ n$, put $E_ m=E\cap L_ m$, and choose a sequence $(m_ j)_{j\ge 1}$ in which every positive integer occurs infinitely often. Since each $L_ m$ is bounded, we can choose radii $R_ j>1$ increasing to $\infty $ such that

in particular $L_{m_ j}$ lies in the ball $B_ j=B(0,R_ j)$. The set $E\cap B_ j$ is pluripolar in $B_ j$, so by theorem 5.47.8 the relative extremal function of its subset $E_{m_ j}$ with respect to $B_ j$ vanishes identically. The competitors of (5.47.4) for $E_{m_ j}$ in $B_ j$ therefore converge to zero almost everywhere on $B_ j$, and by bounded convergence their integrals over $B_ j$ tend to zero. Since a competitor for $E_{m_ j}$ with values in $[-1,0]$ equals $-1$ on $E_{m_ j}$, one of these competitors, which we call $u_ j$, satisfies

The function $\log ^+|z|=\max \{ \tfrac 12\log |z|^2,0\} $ is plurisubharmonic on $\mathbb C^ n$ by proposition 5.10.3, proposition 5.9.5, and so is $\rho _ j$. On $\partial B_ j$ we have $|z|=R_ j>1$, hence $\rho _ j=0$, while the upper limit of $u_ j$ at every point of $\partial B_ j$ is at most $0$. By proposition 5.9.5, the function $V_ j$ equal to $\max \{ u_ j,\rho _ j\} $ on $B_ j$ and to $\rho _ j$ on $\mathbb C^ n\setminus B_ j$ is plurisubharmonic on $\mathbb C^ n$. Since $\rho _ j\le -1=u_ j$ on $E_{m_ j}$, we have $V_ j=-1$ on $E_{m_ j}$. On $B_ j$ both $u_ j$ and $\rho _ j$ are nonpositive, so $V_ j\le 0$ there, while $V_ j=\rho _ j$ outside $B_ j$; hence

In particular $V_ j$ is locally bounded.

Step 2: the sum. Fix $r>0$ and choose $j_0$ with $R_{j_0}>r$. For $j\ge j_0$ the ball $B(0,r)$ lies in $B_ j$, so $V_ j\le 0$ on $B(0,r)$, and since $V_ j\ge u_ j$ there,

Hence the partial sums of $\sum _{j\ge j_0}V_ j$ decrease on $B(0,r)$, and by monotone convergence their limit $W_ r$ satisfies $\int _{B(0,r)}(-W_ r)\, \mathrm dV_{2n}\le \sum _ j2^{-j}<\infty $. Thus $W_ r$ is not identically $-\infty $, and $W_ r\in \mathrm{PSH}(B(0,r))$ by theorem 5.10.1. At every point of $\mathbb C^ n$ all but finitely many terms of the series $V=\sum _ jV_ j$ are nonpositive, so $V$ is well defined with values in $[-\infty ,\infty )$. On $B(0,r)$ it equals $\sum _{j<j_0}V_ j+W_ r$, a sum of finitely many finite plurisubharmonic functions and a plurisubharmonic function; hence $V$ is plurisubharmonic on $B(0,r)$. As $r$ is arbitrary and plurisubharmonicity is a local property, $V\in \mathrm{PSH}(\mathbb C^ n)$.

Let $z\in E$, so that $z\in E_ m$ for some $m$. Since $m=m_ j$ for infinitely many $j$, infinitely many terms $V_ j(z)$ equal $-1$, while all terms with $R_ j>|z|$ are nonpositive. Hence $V(z)=-\infty $. Finally, summing the upper bounds of Step 1 gives $V\le \sum _ j2^{-j}\log ^+|z|=\log ^+|z|$. $\square$