Completion space of $L^1(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$

Question

With Borel measure. Is $L^p(\mathbb{R}^n)$ the completion space of $L^1(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$ with norm $|\cdot |_{p}$?

Answer 1

Functions in $L^{1}\cap L^{p}$ need no to be Borel measurable, taking Borel measure with them is somehow undefined.

Perhaps a proper question would be, for continuous with compactly supported functions $C_{00}$, these functions are Borel measurable, and we know that $\overline{C_{00}}=L^{p}$ under the Lebesgue measure. Since they are already Borel measurable, taking Lebesgue measure to them are the same taking with the Borel measure one, so $L^{p}$ is the completion of $C_{00}$ with Borel measure.