I would like to know if an open set with small capacity also has small measure. To be more specific, I would like to know if the following statement is true:
For each $\epsilon>0$, there exists $\delta>0$ such that if $\Omega\subset\mathbb{R}^n$ is a bounded domain (i.e., open and connected) with $\text{Cap}_p(\Omega) < \delta$, then $|\Omega| < \epsilon$.
Here $1<p<n$, and $\text{Cap}_p(\Omega)$ is the $p$-capacity of the set $\Omega$, i.e., \begin{equation*} \text{Cap}_p (\Omega) = \inf\left\{ \int_{\mathbb{R}^n} |\nabla u|^p\,dx : u \in W^{1,p}(\mathbb{R}^n), u = 1 \text{ on }\Omega \right\}. \end{equation*} Also $|\Omega|$ is the Lebesgue measure of $\Omega$.
Thanks!