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

Question

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

Answer 1

You can choose any simple function that satisfies the condition whether it is supported by finite set or not. In fact, simple function is always bounded and measurable without any condition.