Lemma 7.5.456B45ADBA326

Proof. Let $\eta >0$. By quasi-continuity, applied to the compact set $L$, there is an open set $G\subset D$ with $\operatorname {Cap}_ D(G)<\eta $ such that $f$ is continuous on $L\setminus G$; let $g$ be a continuous extension of $f|_{L\setminus G}$ to $\mathbb {C}^ n$ (lemma 7.5.3); here $f|_{L\setminus G}$ is bounded, being continuous on a compact set. The open set $G\cup (\operatorname{Int}L\cap \{ |g|>\delta /2\} )$ contains $K\cap \{ |f|>\delta \} $, since $K\subset \operatorname{Int}L$ and $g=f$ on $K\setminus G$. Its part outside $G$ is a Borel set contained in $L\cap \{ |f|>\delta /2\} $, because $g=f$ on $L\setminus G$. By monotonicity and subadditivity,

Let $\eta \downarrow 0$. $\square$