$C_c^\infty(\Omega)\subseteq L^p(\Omega)$ for any open $\Omega$? Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$. Can we show that $$C_c^\infty(\Omega)\subseteq L^p(\Omega)\tag 1$$ for all $p\in [1,\infty]$? It's clear that $(1)$ holds if $\Omega$ has finite Lebesgue measure. And it's clear that $(1)$ holds for $p=\infty$.

I'm assuming that  $C^\infty_c(\Omega)$ is the space of $C^\infty$ with compact support contained in $\Omega$. If that's the case, then you don't care whether or not $\Omega$ has finite measure, since
$$
\int_\Omega |u|^pdx\leq |spt(u)|\cdot\max |u|^p<\infty.
$$
where $spt(u)$ is the support of $u$, that is defined as
$$
spt(u)=\overline{\{x\in \Omega : u(x)\neq 0\}}.
$$
Since the support is compact, it is boudned, so $|spt(u)| < \infty$.
