Measurable Function is Simple Function iff Finite Image Set

Theorem
Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $f: X \to \R$ be a real-valued function.

Then $f$ is a simple function iff its image is finite:


 * $\# \operatorname{Im} \left({f}\right) < \infty$

Corollary
Every simple function $f: X \to \R$ has a standard representation.