# A question about openness theorem

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. Degenerate Complex Monge-Ampere Equations, Vincent Guedj and Ahmed Zeriahi. This is not a trivial result, however in the context of $\varphi\left(0\right)=-\infty$ in your question, it's trivial.