L^{r}(\Omega,\mu) \cap L^{p}(\Omega,\nu)$ there exists some $\phi$ in $\hbox{Lip}_c (\Omega)$ such that $$\|f -\phi\|_{L^r(\Omega,\mu)} \leq \epsilon \quad \text{and} \quad \|f-\phi\|_{L^p(\Omega,\nu)} \leq \epsilon.$$

By Urysohn’s lemma $C_c (\Omega)$ is dense in $L^r(\Omega,\mu)$ and in $L^p(\Omega,\nu)$.

To prove the lemma it is sufficient to check that for every measurable set $A$ the characteristic function $\mathbf{1}_A$ can be approximated both in $L^r(\Omega,\mu)$ and in $L^p(\Omega,\nu)$ by functions in $\hbox{Lip}_c (\Omega)$. The regularity of the measures ensures that there exists a sequence $\{K_n\}_n$ of compact sets and a sequence $\{U_n\}_n$ of open sets such that $ K_n \subset A \subset U_n $, with $$\mu(U_n \thicksim K_n) \leq \frac{1}{n}\   \text{and} \ 
\nu(U_n \thicksim K_n) \leq \frac{1}{n}.$$ Without loss of generality we can assume that $U_n \subset \Omega$. Moreover, since the space is locally compact, for every $n$ there exists an open precompact set $V_n$ such that $$K_n \subset V_n \subset \overline{V}_n \subset U_n.$$ Introduce the sequence of functions $\{\psi_n\}_n$ given for each $n$ by $$\psi_n := \mathbf{1}_{K_n} : K_n \cup (\Omega \thicksim V_n)\rightarrow [0,1] ,$$ and denote by $\tilde{\psi}_n$ the extension of $\psi_n$ to all $\Omega$ defined as $$\tilde{\psi}_n(x) := \sup_{y \in  K_n \cup (\Omega \thicksim V_n) } \left(  \psi_n(y) - L_n\ d(x,y) \right) ,$$ 