  We consider integral inequalities in the sense of Choquet with respect to the
Hausdorff content $\mathcal{H}_\infty^{\delta}$. In particular, if $\Omega$ is
a bounded John domain in $\mathbb{R}^n$, $n\geq 2$, and $0 <\delta \le n$, we
prove that the corresponding $(\delta p/(\delta -p),p)$-Poincar\'e-Sobolev
inequalities hold for all continuously differentiable functions defined on
$\Omega$ whenever $\delta /n < p < \delta$. We prove also that the
$(p,p)$-Poincar\'e inequality is valid for all $p>\delta /n$.
