Lusin's theorem says that in a finite measure space, given a measurable function $\varphi$, for every $\varepsilon \gt 0$ there exists a continuous function $g$ such that $$ \mu\left(\{x : \varphi(x)\neq g(x)\}\right)\lt \varepsilon.$$
How can I use this to show that every $f\in L^p$ can be approximated in $L^p$ by continuous functions of compact support?