Query in the proof of $\overline{C_c(X)}=L^p(\mu)$ of Rudin's Real & Complex Analysis (Theorem 3.14)

Question

I am reading the proof of the Theorem 3.14 from Rudin's Real & Complex Analysis book which says

For $1\le p<\infty, C_c(X)$ is dense in $L^p(\mu)$ where $X$ is locally compact hausdorff space.

In the proof, he used Lusin's Theorem on a simple function $s$. But the statement of Lusin's Theorem says (Theorem 2.24)- To apply this theorem on a simple function $s$, we should have a set $A$ with finite measure (i.e. $\mu(A)<\infty$) such that $s$ is zero outside $A$. Now if $s=\sum\limits_{i=1}^n \alpha_i \chi_{A_i}$ where $\alpha_i\in\Bbb{C}\setminus\{0\}$ and $A_i$'s are disjoint measurable sets. So, $s$ is zero outside $\bigsqcup\limits_{i=1}^n A_i$ . But it's measure $\mu(\bigsqcup\limits_{i=1}^n A_i)=\sum\limits_{i=1}^n \mu(A_i)$ may be $\infty$. Then how can we apply Lusin's Theorem here? Can anyone clarify my doubt? Thanks for help in advance.

Answer 1

Replace $f$ by $s$ and take $A=\{x: s(x) \neq 0\}$ in Lusin's Theorem.