Bounded, measurable and supported on a set of finite measure function

Suppose f is a bounded and measurable function on R and supported on a set of finite measure. Prove that for every $\epsilon \gt 0$ there exists a simple function $s$ such that $\int |f-s|dx$ $\lt \epsilon$

I have question: Can I let this simple function be supported on a set of finite measure? Then this simple function will be bounded and measurable and supported, then I can use the linearity of integral and the definition of the integral of this kind of functions to prove that.

I am wondering if I am right. I really appreciate your help if you would like to give me another proof.

Thanks