Proposition 5.40.3630365A8F8B8

Proof. The constant $\min _ Xf$ is a competitor, so $p=P_\omega (f)$ is bounded. The local envelope theorem gives $p\leq f$ quasi-everywhere, and continuity of $f$ makes this inequality hold everywhere. By theorem 5.32.1, there are smooth $\omega $-plurisubharmonic functions $v_ j\downarrow p$. Fix $\epsilon >0$. The continuous functions $v_ j-f$ decrease to a function that is at most zero, so $v_ j\leq f+\epsilon $ everywhere for all large $j$; otherwise the decreasing compact sets $\{ v_ j-f\geq \epsilon \} $ would have nonempty intersection. Hence $v_ j-\epsilon $ is a competitor, and

Thus $v_ j\to p$ uniformly, which proves continuity. The last inequality follows from the monotonicity of $P_\omega $ in the obstacle and the identity $P_\omega (g+c)=P_\omega (g)+c$ for constants $c$, applied with $c=\| f-g\| _\infty $. $\square$