$D(\mathbb{R}^n) \subset L^{p}(\mathbb{R}^n)$

Question

I have started reading distribution theory and there author state that $\mathcal{D}(\mathbb{R}^n) \subset L^{p}(\mathbb{R}^n)$ where $\mathcal{D}(\mathbb{R}^n)$ is the set of all complex valued infinitely differentiable function with compact support and $L^{p}(\mathbb{R}^n)$ have the usual meaning.

I cant prove the result. Please help.

Answer 1

Remember that any continuous function is Borel measurable. $\int |f|^{p} \,\mathrm{d}m =\int_K |f|^{p} \,\mathrm{d}m \leq C^p m(K)<\infty$ where $K$ is the support of $f$ and $C$ is the supremum of $|f|$ ($m$ being Lebesgue measure on $\mathbb R^{n}$).

Answer 2

Hint: $f \in C_c^\infty(\mathbb{R}^n)$ is bounded (why?), and so is $|f|^p$. Also, there is some compact $K$ such that $|f|^p = 0$ for $x \not\in K$ (why?). Combine both to prove the above result.