$C_c^\infty(\Omega)\subseteq L^p(\Omega)$ 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 $p=\infty$.

Well yeah it has compact support, hence outside of a compact set $K$ is zero. It's continuos on a bounded, closed set hence its integral must be smaller than $M\cdot m(K)$, where $M$ is the maximum of the function on $K$ and $m$ is the lebesgue measure 
