Lemma 13.15.3357B2DE7779E

Proof. Put $v_ t=\max (u,-t-1)+t+1$. It is $\omega $-plurisubharmonic and takes values in $[0,t+1]$, and by plurifine locality its Monge–Ampère measure equals $\mu _ u$ on $\{ u\geq -t\} \subset \{ u>-t-1\} $. The first estimate therefore follows from proposition 13.15.2, after enlarging the constant. For the second, $v_ t/(1+t)$ is a test function for the capacity, and positivity gives

Restricting this inequality to $E\cap \{ u\geq -t\} $ proves the capacity bound. $\square$