PomeProposition 5.47.25.6EBEFACB2203 Let $\Omega \subset \mathbb C^ n$ be open, let $\varphi ,\psi \in \mathrm{PSH}(\Omega )$, and put $P=\{ \varphi =\psi =-\infty \} $. Then $P$ is plurifine closed and pluripolar, and the function $\varphi -\psi $ is continuous from the plurifine open set $\Omega \setminus P$, with the plurifine topology, to $[-\infty ,\infty ]$. The same holds for $\varphi -\psi +b$, for every continuous real function $b$ on $\Omega $.
5.47.24 Common poles and quasi-extrema6AAE49F7183F
Let $\varphi $ and $\psi $ be plurisubharmonic functions on an open set $\Omega \subset \mathbb C^ n$. Their difference $\varphi -\psi $ is defined, with values in $[-\infty ,\infty ]$, outside the common pole set $\{ \varphi =\psi =-\infty \} $, on which it would be the meaningless expression $-\infty -(-\infty )$. We show that $\varphi -\psi $ is plurifinely continuous off the common pole set, and that its supremum and infimum over a plurifine open set do not change when an arbitrary pluripolar set containing the common poles is removed.
Proof. The function $\varphi $ is plurifinely continuous with values in $[-\infty ,\infty )$, and $\{ -\infty \} $ is closed in $[-\infty ,\infty )$; hence the pole set $\{ \varphi =-\infty \} $ is plurifine closed. The same holds for $\psi $, so $P$ is plurifine closed. It is pluripolar because it lies in the pole set of $\varphi $, which is pluripolar by definition 5.47.2, applied with small balls in $\Omega $ as neighborhoods.
The subtraction $(s,t)\mapsto s-t$ is continuous from $[-\infty ,\infty )^2\setminus \{ (-\infty ,-\infty )\} $ to $[-\infty ,\infty ]$. At points where $s$ and $t$ are finite, this is the continuity of subtraction on $\mathbb R^2$. If $s=-\infty $ and $t$ is finite, then for every $M>0$ we have $s'-t'<-M$ whenever $t'>t-1$ and $s'<t-1-M$; this is continuity with the value $-\infty $. Symmetrically, if $s$ is finite and $t=-\infty $, then $s'-t'>M$ whenever $s'>s-1$ and $t'<s-1-M$. Since $z\mapsto (\varphi (z),\psi (z))$ is plurifinely continuous on $\Omega \setminus P$ with values in this domain, $\varphi -\psi $ is plurifinely continuous on $\Omega \setminus P$. Finally, $b$ is continuous for the Euclidean topology, hence for the plurifine topology, and addition is continuous from $[-\infty ,\infty ]\times \mathbb R$ to $[-\infty ,\infty ]$. $\square$
Definition 5.47.26.FF965EAEB1A3 Let $X$ be a complex manifold, let $W\subset X$ be a nonempty plurifine open set, let $Z\subset X$ be pluripolar, and let $f\colon W\setminus Z\to [-\infty ,\infty ]$. The quasi-supremum and the quasi-infimum of $f$ on $W$ are
\[ \operatorname *{q-sup}_ W f =\inf _{N}\, \sup _{W\setminus N}f, \qquad \operatorname *{q-inf}_ W f =\sup _{N}\, \inf _{W\setminus N}f, \]
where $N$ ranges over the pluripolar subsets of $X$ containing $Z$, and all suprema and infima are taken in $[-\infty ,\infty ]$.
The sets $W\setminus N$ in this definition are nonempty: by proposition 5.47.16(1), $W$ contains a Borel set of positive Lebesgue measure in some chart, whereas $N$ is Lebesgue null in every chart.
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 $Z$. $\square$
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