How to show that $C^\infty_0$ is not dense in $L^p_{weak} (\mathbb{R}^n)$? Let $C^\infty_0$ denote the  smooth, compactly supported functions on $\mathbb{R}^n$. Let $L^p_{\mathrm{weak}}(\mathbb{R}^n)$ denote the space of all functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ which satisfy
$$
 \Big| \big\{x \in \mathbb{R}^n : |f(x)| > \lambda \big\} \Big| \lesssim \lambda^{-p}
$$
for all $\lambda > 0$. This space when endowed with the quasi-norm
$$
\Vert f\Vert_{L^p_{\mathrm{weak}}}^* : = \sup_{\lambda > 0} \lambda \Big| \big\{x \in \mathbb{R}^n : |f(x)| > \lambda \big\} \Big|^{1/p}
$$
is a quasi-Banach space. How do you show that $C^\infty_0$ is not dense in $L^p_{\mathrm{weak}} (\mathbb{R}^n)$ for $1< p < \infty$? This question was inspired by a real analysis qualifying problem, and I know that considering the function $f(x) = |x|^{-n/p}$ is important.

The point is that the function $f\colon x\mapsto \left\lvert x\right\rvert^{-n/p}$ belongs to $\mathbb L^p_{\operatorname{weak}}\left(\mathbb R^n\right)$ and for each fixed $R$, 
$$
\left\lVert f-f\cdot\mathbf 1_{[-R,R]^n}\right\rVert_{\mathbb L^p_{\operatorname{weak}}}\geqslant c
$$
where $c$ depends only on $n$. Therefore, for any function $g$ with compact support, $\left\lVert f-g\right\rVert_{\mathbb L^p_{\operatorname{weak}}}\geqslant c$.
