PomeProposition 7.5.6.9542ED69E1C1 Let $u_ j$ be locally bounded plurisubharmonic functions on $D$ decreasing to a locally bounded function $u$. Then $u_ j\to u$ locally in capacity. Moreover, a net of real Borel functions on $D$ which is uniformly bounded on each compact subset of $D$ and converges locally in capacity to a locally bounded real Borel function also converges in $L^1_{\mathrm{loc}}(D)$.
Proposition 7.5.69542ED69E1C1
Proof. Let $K\subset D$ be compact and $\delta >0$. Since $u_ j\ge u$, the sets $\{ u_ j<u-\delta \} $ are empty, and we must show that $\operatorname {Cap}^*_ D(K\cap \{ u_ j-u>\delta \} )\to 0$. By theorem 5.10.1, $u$ is plurisubharmonic. Covering $K$ by finitely many balls $B(x,r)$ with $\overline{B(x,4r)}\subset D$ and using subadditivity of outer capacity, we may assume that $K\subset \overline{B(x,r)}$ for such a ball; put $L=\overline{B(x,3r/2)}$, $B=B(x,3r)$ and $B_1=B(x,4r)$.
We first make the functions coincide near $\partial B$. Since $u\le u_ j\le u_1$, all these functions are bounded on $B_1$ uniformly in $j$, so there are constants $c$ and $N\ge 1$ with $-N\le u_ j-c,\, u-c\le -1$ on $B_1$. Put $\psi =|z-x|^2-9r^2$, choose $\delta _0>0$ with $L\subset \{ \psi <-\delta _0\} $, and let $\hat u_ j=\max \{ u_ j-c,A\psi \} $ and $\hat u=\max \{ u-c,A\psi \} $, with $A=N/\delta _0$, be the functions (7.3.2) on $B_1$, with $\delta _0$ in place of $\delta $. They are bounded and plurisubharmonic on $B_1$, $\hat u_ j\downarrow \hat u$, $\hat u_ j-\hat u=u_ j-u$ on $L$, and $\hat u_ j=\hat u$ on a collar of $\partial B$. By (7.4.6), applied with $u=\hat u$ and $h=\hat u_ j$,
\[ \operatorname {Cap}_ D\bigl (B\cap \{ \hat u_ j-\hat u>\delta /2\} \bigr ) \le (n+1)!\, (2/\delta )^{n+1}\int _ B(\hat u_ j-\hat u)(\mathrm{dd}^{\mathrm{c}}\hat u)^ n, \]
and the right-hand side tends to zero by dominated convergence, since $(\mathrm{dd}^{\mathrm{c}}\hat u)^ n$ has finite mass on $B$ and $0\le \hat u_ j-\hat u\le \hat u_1-\hat u$ decreases to zero. The real function $w_ j=u_ j-u$ is quasi-continuous on $D$, as a difference of two real quasi-continuous functions (theorem 7.5.2 and the remark following definition 7.5.1), and $w_ j=\hat u_ j-\hat u$ on $L$. Hence lemma 7.5.4 gives
\[ \operatorname {Cap}^*_ D(K\cap \{ w_ j>\delta \} )\le \operatorname {Cap}_ D(L\cap \{ w_ j>\delta /2\} )\le \operatorname {Cap}_ D\bigl (B\cap \{ \hat u_ j-\hat u>\delta /2\} \bigr )\longrightarrow 0. \]
For the second assertion, let $f_ a\to f$ be as in the statement, let $K\subset D$ be compact, and let $M$ bound $|f_ a|$ and $|f|$ on $K$. Splitting $K$ into the Borel set where $|f_ a-f|>\delta $ and its complement, and using proposition 7.4.3(3) together with $\operatorname {Cap}_ D\le \operatorname {Cap}^*_ D$ on Borel sets, we get
\[ \int _ K|f_ a-f|\, \mathrm{d}V_{2n} \le 2MC_ D\operatorname {Cap}_ D^*(K\cap \{ |f_ a-f|>\delta \} ) +\delta |K|. \]
Taking the limit along the net and then letting $\delta \downarrow 0$ proves $L^1$ convergence on $K$. $\square$
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