Corollary 5.47.27.D6E89A0A4867 Let $\Omega \subset \mathbb C^ n$ be open, let $\varphi ,\psi \in \mathrm{PSH}(\Omega )$, let $b$ be a continuous real function on $\Omega $, and let $f=\varphi -\psi +b$ on $\Omega \setminus P$, where $P=\{ \varphi =\psi =-\infty \} $. Let $Z\subset \Omega $ be a pluripolar set containing $P$, and let $W\subset \Omega $ be a nonempty plurifine open set. Then
\[ \operatorname *{q-sup}_ W f=\sup _{W\setminus Z}f, \qquad \operatorname *{q-inf}_ W f=\inf _{W\setminus Z}f. \]
In particular, the right-hand sides do not depend on the choice of $Z$.
Proof.
The set $W'=W\setminus P$ is plurifine open, and $f$ is plurifinely continuous on $W'$, by proposition 5.47.25. Let $N$ be a pluripolar set containing $P$. We claim that $\sup _{W'\setminus N}f=\sup _{W'}f$. The inequality $\le $ holds because $W'\setminus N\subset W'$. Conversely, if $s\in \mathbb R$ and $s<\sup _{W'}f$, then $\{ z\in W':f(z)>s\} $ is a nonempty plurifine open set. It meets $\Omega \setminus N$, which is plurifine dense by proposition 5.47.16(2), and hence $\sup _{W'\setminus N}f>s$. This proves the claim, and the same argument with the sets $\{ f<s\} $ gives $\inf _{W'\setminus N}f=\inf _{W'}f$.
Since $W\setminus N=W'\setminus N$ whenever $N\supset P$, every pluripolar set $N\supset Z$ gives $\sup _{W\setminus N}f=\sup _{W\setminus P}f$; taking $N=Z$, we also get $\sup _{W\setminus Z}f=\sup _{W\setminus P}f$. Hence $\operatorname *{q-sup}_ Wf=\sup _{W\setminus Z}f=\sup _{W\setminus P}f$, and similarly $\operatorname *{q-inf}_ Wf=\inf _{W\setminus Z}f=\inf _{W\setminus P}f$. The common values do not involve $Z$.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.