Proposition 16.25.2467197AADC50

Proof. We begin with the final identities. A competitor for $P_\theta [u](0)$ is a nonpositive potential more singular than $u$, exactly as in definition 16.21.1; and every potential is more singular than $V_\theta $, by minimality, so the families defining $P_\theta [V_\theta ](f)$ and $P_\theta (f)$ coincide.

(1) The maximum of two members of the family in definition 16.25.1 is again a member. By theorem 5.10.2, applied on finitely many coordinate balls on which $\theta $ has smooth local potentials, there are members $v_1,v_2,\ldots $ with $(\sup _ kv_ k)^*=P_\theta [u](f)$, and the maxima $w_ k=\max \{ v_1,\ldots ,v_ k\} $ form an increasing sequence of members with the same regularized supremum. By corollary 5.47.11, $\sup _ kw_ k=P_\theta [u](f)$ outside a pluripolar set, and each $w_ k$ lies below $f$ outside a pluripolar set; since countable unions of pluripolar sets are pluripolar (theorem 5.47.8), $P_\theta [u](f)\leq f$ quasi-everywhere. (The envelope itself need not be more singular than $u$; see example 16.22.12.) The other assertions follow from the corresponding properties of the families of competitors.

(2) A competitor $h$ for the obstacle $\min \{ u+k,f\} $ satisfies $h\leq u+k$ quasi-everywhere, hence everywhere, and $h\leq f$ quasi-everywhere; so it belongs to the family of definition 16.25.1. In particular these competitors are uniformly bounded above, $P_\theta (u+k,f)$ is defined, and when it is a potential, it belongs to the family. Conversely, every member $v$ of the family satisfies $v\leq u+k$ for all sufficiently large integers $k$, hence $v\leq \min \{ u+k,f\} $ quasi-everywhere and $v\leq P_\theta (u+k,f)$. Thus a fixed member shows that the rooftop envelopes, which increase with $k$, are potentials for all large $k$, and the family and the rooftop envelopes have the same supremum. This proves the representation.

(3) Normalize $u\leq 0$. If $|f|\leq C$ quasi-everywhere, then $u-C$ is a competitor, and for large $k$ each rooftop envelope $P_\theta (u+k,f)$ lies between $u-C$ and $u+k$, so it has the singularity type of $u$ and mass $\int _ X\langle \theta _ u^ n\rangle $. By (2) and corollary 13.8.9, $P_\theta [u](f)$ also has this mass. Comparing $f$ with the constants $-C$ and $C$, and using (1) together with $P_\theta [u](0)=P_\theta [u]$, gives the two bounds. $\square$