PomeProposition 8.13.3.CCDE5B297235 If $u,v$ are bounded admissible and $w=\max \{ u,v\} $, then
\[ \int _{\{ u<v\} }\mathrm d\mu _ r(v) \leq \int _{\{ u<v\} }\mathrm d\mu _ r(u) +\int _ X\mathrm d\mu _ r(w)-\int _ X\mathrm d\mu _ r(u). \]
Proposition 8.13.3CCDE5B297235
Proof. Set $E=\{ u<v\} $. The contact-set version of 8.12.2 gives
\[ \mu _ r(w)\geq \mathbf1_{X\setminus E}\mu _ r(u) +\mathbf1_ E\mu _ r(v). \]
All these measures are finite and positive. Integrating on $X$ and using $\mu _ r(u)(X\setminus E)=\mu _ r(u)(X)-\mu _ r(u)(E)$ gives exactly the assertion. No boundary or closed-background hypothesis enters this calculation. $\square$
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