$C(A)$ is $\|\cdot\|_2$-dense in $\ell_2(A)$

Question

Let $A \neq \varnothing$ and $\cal {F}$$(A) = \{F \subset A \mid F$ is finite$\}$.

Define $\ell_2 (A) =L^2(A, 2^A, \mu_C)$, with $\mu_C$ the counting measure.

Let $C(A) = \{f: A \to \Bbb C, \exists \ F \in \cal F$$(A) \mid f = 0$ on $A \setminus F \}$.

The following holds:

$C(A)$ is $\|\cdot\|_2$-dense in $\ell_2(A)$.

Where can I find a proof of this claim?

Background: it was given to us in class as an admitted theorem. I do not think that the proof is very advanced, so I would like to see one.

Answer 1

Hint: If $f:A \to \Bbb C$ is in $\ell_2(A)$ (that is, if $|f|^2$ has a convergent integral with respect to counting measure), then it is zero on all but countably many points.

Hence, we can apply the proof from the case of $A=\Bbb N$.