where $L_n = 1 / \text{ dist } ( K_n , \Omega \thicksim V_n)$. Such functions are Lipschitz on $\Omega$, with $\tilde{\psi}_n = \psi_n$ on $ K_n \cup (\Omega \thicksim V_n)$, and with $\tilde{\psi}_n \leq 1$. Finally, define $$\phi_n = \max \{0, \tilde{\psi}_n \},$$ and note that $\phi_n \in \hbox{Lip}_c (\Omega)$. Then it is easy to see that $$\int_{\Omega} \left| \phi_n(x) - \mathbf{1}_A(x) \right|^{r} d \mu(x) \leq 2^{r} \mu(U_n \thicksim K_n) \leq \frac{2^{r}}{n} ,$$ and similarly $$\int_{\Omega} \left| \phi_n(x) - \mathbf{1}_A(x) \right|^{p} d \nu(x) \leq \frac{2^{p}}{n}.$$

Proof of Theorem \[main2\] {#main}
==========================

In this section we prove Theorem \[main2\], our main result. To prove such a theorem, we will need Theorem \[theorem:Lions\], which in turn requires Lemma \[lemma:rhol\], Lemma \[prod\] and Lemma \[BL\]. We start with Lemma \[lemma:rhol\]:

\[lemma:rhol\] (Reverse Hölder) Let $\Omega \subset X$ be an open precompact subset of the metric space $(X, d)$, and let $\mu$ and $\nu$ be Radon measures on $\Omega$. Assume that $1 \leq p < r$. If there exists a positve real number $C$ such that for every Lipshitz $\phi$ with compact support $$\label{hypothesis}
\| \phi \|_{L^r(\Omega,\mu)} \leq C \| \phi \|_{L^{p}(\Omega,\nu)},$$ then there exists a countable set of points $\{x_i \}_{i \in 