simple function is dense in L^p space

Question

I've been looking at the proof of 'simple function is dense in $L^p$ space in the Rudin's book 'Real and complex analysis(RCA)', Page 69, theorem 3.13.

My question is in the assumption, there is a 'finite support simple function'. But the proof does not prove the simple function has finite support. Rudin might think this is obvious, but I'm not.

How could we verify this?

Answer 1

By definition a simple function $s$ has a finite range. In particular, if non-negative it has a minimum value $m>0$ on its support $\Omega$. If $L^p$-integrable (as in Rudin) we must have $$m^p\mu(\Omega)\leq \int s^p <+\infty$$ so the support must be finite.