Proposition 16.28.7F0B070E89AE8

Proof. Since $\varphi =P_\theta [\varphi ]\leq 0$, the function $\varphi $ is a competitor for $P_\theta (\varphi +1,0)$. This rooftop is also a competitor for $P_\theta [\varphi ]=\varphi $, being at most $\varphi +1$ and at most zero, so it equals $\varphi $. Now apply proposition 16.28.6 with $u=\varphi +1$ and $v=0$. The contact set $\{ \varphi =\varphi +1<0\} $ is contained in the pole set of $\varphi $, which the extended measure does not charge. Only contact with zero contributes, and this gives the asserted bound. $\square$