Showing or refuting $L^1(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$ for $1\leq p\leq \infty$?

Question

$L^1(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$ for $1\leq p\leq \infty$?

Answer 1

This is false for $p=\infty$: take $f(x)\equiv 1$. If $f_n \in L^{1}$ and $||f-f_n\|_{\infty} \to 0$ then $|f_n| \geq \frac 1 2$ for large enough $n$ so $f_n$ cannot be integrable. Also the result is obvious for $p=1$.

For $1<p<\infty$ just take $f_n=fI_{\{x:|x| \leq n\}}$. These functions are in $L^{1}\cap L^{p}$ and tend to $f$ in $L^{p}$ by DCT.

To prove that $f_n \in L^{1}$ use Holder's inequality: you get $\|f_n\|_1\leq \|f\|_p (\mu(\{x:|x| \leq n\})^{1/q}$ where $\frac 1 p+ \frac 1 q=1$.