Lemma 13.19.450A87754E163

Proof. Set $p_ C=P_\theta (u+C,w)$ for $C\geq 0$ and $p=P_\theta [u](w)$. The envelopes $p_ C$ increase with $C$, lie between $p_0$ and $w$, and converge to $p$ in capacity by lemma 5.29.19. By the contact theorem, $\mu _{p_ C}$ is carried by $\{ p_ C=\min (u+C,w)\} $. Assigning the points where $u+C=w$ to the second term, ordered contact gives

The first measure has total mass at most

because $\mu _ u$ is finite and non-pluripolar. Testing against nonnegative continuous functions and applying the lower semicontinuity of lemma 13.14.8 gives $\mu _ p\leq \mu _ w$.

The bounded quasi-continuous weight $\chi =\min (1,(w-p)_+)$, defined arbitrarily on the pluripolar set where $w-p$ is undefined, vanishes on $\{ p_ C=w\} $, since $p_ C\leq p\leq w$. Thus $\int _ X\chi \, \mu _{p_ C}\leq \varepsilon _ C$. Weighted lower semicontinuity gives $\int _ X\chi \, \mu _ p=0$, so $\mu _ p$ is carried by $\{ p=w\} $. Together with $\mu _ p\leq \mu _ w$, this proves the lemma. $\square$