Lemma 16.28.68F644817639FD

Proof. Since $\varphi \leq 0$, the potential $u$ lies below the obstacle $u-\varphi $ away from the pole set of $\varphi $, so it is a competitor. Hence the envelope $w$ is not identically $-\infty $. It is at most zero, since $u-\varphi \leq 0$, and it satisfies $w+\varphi \leq u$ off the poles, hence everywhere by lemma 5.24.3. For $b\geq 1$, let $h_ b=Q_ b(u,\varphi )$. Then, off the poles,

Since $\theta \geq 0$, the function $b^{-1}h_ b$ is $\theta $-plurisubharmonic, and so it is a competitor for $w$. Hence $h_ b\leq bw$, so $\mathcal A_ b(w,0)$ is nonempty for every $b$. Since also $w\leq 0$, this means exactly that $w\in \mathcal E_0$. $\square$