I want to prove that for each $1\leq p < \infty$,
$C_c^\infty (U) = \{f|_U : f \in C_c^\infty(\mathbb R^n), \text{supp }f \subseteq U \}$ is dense in $\mathcal L^p(U, \mathcal B(U), \lambda_n|_U)$.
I know that $C_c^\infty(\mathbb R^n) $ is dense in $\mathcal L^p(\mathbb R^n)$. But I am not sure in how far that easily generalizes so I tried to make prove it from scratch:
Let $U\subseteq \mathbb R^n$ open, $g\in \mathcal L^p(U)$, $K_m \uparrow U$ be an increasing sequence of compact sets with $\bigcup_{n\in \mathbb N} K_n = U$. (Such $K_m$ always exist since every open subset of $\mathbb R^n$ can be written as a union of at most countably many open sets of the form $\prod_i]a_i, b_i[$, then just take $\prod_i [a_i +1/n, b_i - 1/n]$.
Then $(g 1_{K_m})_m$ are all compactly supported with supp $f \subseteq U$ and $g1_{K_m} \longrightarrow g$ as $m\to \infty$.
Now let $f\in C_c^\infty(\mathbb R^n)$ with supp $f\subseteq U$ and define the mollifier $f_\epsilon(x) := \epsilon^{-n } f(x/\epsilon) $.
Then, for every $\epsilon > 0$, $f_\epsilon * 1_{K_m}g$ is smooth with support in $U$ (is this correct? How else do I have to pick my functions so that this works?).
We get $$f_\epsilon * g1_{K_m} \to g1_{K_m} \to g,$$ as desired. Is this proof correct and/or are there easier ways to to this knowing $C_c^\infty(\mathbb R^n)$ is dense?
I think it's actually much easier to obtain, since every function in $L^p(U)$ is, if you extend it by zero, also in $L^p(\mathbb{R}^n)$. Therefore, let $f \in L^p(U)$ and $\bar{f} \in L^p(\mathbb{R}^n)$ be its extension. Since $C_c^\infty(\mathbb{R}^n)$ is dense, there is a sequence $f_n \in C_c^\infty(\mathbb{R}^n)$ converging to $\bar{f}$ in $L^p$-norm. The only thing we have to take care of is that the supports of $f_n$ could be outside of $U$. So define $g_n = f_n|_U \in C_c^\infty(U)$. Edit: To clarify this further, this means that $g_n: U \to \mathbb{C}, g_n(x)=f_n(x) \ \text{for} \ x \in U$.
Now it is easy to see that $g_n$ actually converges to $f$ in the norm of $L^p(U)$, like this: $$ \int_U |g_n(x)-f(x)|^p dx = \int_U |f_n(x) -f(x)|^p dx \leq \int_{\mathbb{R}^n} |f_n(x) - \bar{f}(x)|^p dx $$ and the last integral goes to zero.