The openness theorem says that:
If $\varphi$ be a negative plurisubharmonic function in the unit ball $B(0,1)$ in $\mathbb{C}^{n}$ satisfying $$ \intop_{B(0,1)}e^{-\varphi}<\infty, $$ then there exist $p>1$ and $r>0$ such that $$ \intop_{B(0,r)}e^{-p\varphi}<\infty. $$
My question is: is the above theorem trivial if we don't assume $\varphi\left(0\right)=-\infty$? That is, if, say, $\varphi\left(0\right)=-1$ then it's trivial to prove the above theorem.
It's known that if $h$ is PSH near $0$ and $e^{-h}$ is not locally integrable at $0$ then $h(0)=-\infty$, see e.g. [1]. This is not a trivial result: however, in the context of $\varphi\left(0\right)=-\infty$ in your question, it's trivial.
Reference
[1] Vincent Guedj, Ahmed Zeriahi, Degenerate complex Monge-Ampère equations, (English) EMS Tracts in Mathematics 26. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-167-5/pbk; 978-3-03719-667-0/ebook), pp. xxiv+472 (2017), MR3617346, Zbl 1373.32001.
