Function Simple iff Positive and Negative Parts Simple

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

Let $g: X \to \overline{\R}$ be an extended real-valued function.

Then $g$ is a simple function iff its positive part $g^+$ and negative part $g^-$ are simple functions.

Necessary Condition
Suppose $g$ is a simple function.

By Positive Part of Simple Function is Simple Function, so is $g^+$.

By Negative Part of Simple Function is Simple Function, so is $g^-$.

Sufficient Condition
Suppose $g^+$ and $g^-$ are simple functions.

From Difference of Positive and Negative Parts:


 * $g = g^+ - g^-$

Hence $g$ is simple, by Pointwise Difference of Simple Functions is Simple Function.