Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$. We denote $d\left(x\right)$ the distance from $x$ to the boundary of $\Omega$, that is $$ d\left(x\right):=\inf\left\{ \left\Vert x-y\right\Vert :y\in\partial\Omega\right\} . $$ In the book of A.Kufner: Weighted Sobolev spaces, page 50, he claimed that (without any explanation): there always exists $\varepsilon_{0}>0$ such that $$ \int_{\Omega}d^{-\varepsilon_{0}}\left(x\right)dx<\infty. $$

I suspect this is not true. I think at least we need some assumption on the regularity of $\partial\Omega$, for example $\partial\Omega$ is Lipschitz.

**My question**: is the above statement is true for any
open bounded domain?