Corollary 16.20.46EFE0C8DBCE3

Proof. A constant shift of $V_\theta $ lies below the bounded function $f$, so the competitor family of $P_\theta (f)$ is nonempty, and it is bounded above by $\max _ Xf$. Since $f$ is continuous, $u=P_\theta (f)$ satisfies $u\leq f$ everywhere, by proposition 16.11.4. Theorem 16.20.2, with all $\theta _ j=\theta $, $u_ j=u$ and $f_ j=f$, identifies the restriction of $\langle (\theta +\mathrm{dd}^{\mathrm{c}}u)^ n\rangle $ to $\{ u=f\} $ with that of $(\theta +\mathrm{dd}^{\mathrm{c}}f)^ n$. By theorem 16.13.4, applied to the continuous and hence quasi-continuous obstacle $f$, the measure $\langle (\theta +\mathrm{dd}^{\mathrm{c}}u)^ n\rangle $ is carried by $\{ u=f\} $. This proves the first identity. For $f=0$ the envelope is $V_\theta $ by definition, which gives the second. $\square$