Let $X$ be a locally compact Hausdorff space, $\mu$ a Radon measure on $X$, and $\mathscr{P}(f) = \{1\leq p < \infty : f\in \mathscr{L}^p(\mu)\}$.
It's known that $C_c(X)$ is dense in $\mathscr{L}^p(\mu)$. However, can we approximate any measurable $f$ simultaneously in all $\mathscr{L}^p(\mu)$?
That is, given a measurable $f$, can we always find a sequence $f_n\in C_c(X)$, such that $f_n\to f$ in $\mathscr{L}^p(\mu)$ for all $p\in\mathscr{P}(f)$ ?