Proposition 5.47.256EBEFACB2203

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 $[-\infty ,\infty ]$. $\square$