Let $X$ be a locally compact (Hausdorff) space and let $\mu$ be a (Radon) measure. Can compactly supported functions $X\to\mathbb{C}$ be approximated by simple functions w.r.t. the norm $$\|f\|_{1}:=\int_{X}|f(x)| \ \text{d}\mu(x).$$ A simple function is a finite linear combination of functions of the form $1_{A}$, where $A$ is measurable with $\mu(A)<\infty$.
I know that simple functions are dense in $L^{1}(X)$ (i.e. the completion of $C_{c}(X)$ w.r.t. $\|\cdot\|_{1}$). But I am looking for a direct proof of the above statement. Any suggestions would be appreciated!
Yes. First consider the real and imaginary part, and for each of that consider the positive and negative part. By doing that, it is enough to show this statement for positive functions. Now for a positive measurable function we have that it can be approached (pointwise) from a increasing family of simple functions. Applying monotone convergence theorem gives the result