Negative Part of Simple Function is Simple Function

Theorem
Let $\struct {X, \Sigma}$ be a measurable space.

Let $f: X \to \R$ be a simple function.

Then $f^-: X \to \R$, the negative part of $f$ is also a simple function.

Proof
Let $f$ have the following standard representation:


 * $f = \ds \sum_{i \mathop = 0}^n a_i \chi_{E_i}$

Then we see that $f^-$ must satisfy:


 * $f^- = \ds \sum_{i \mathop = 0}^n \min \set {a_i, 0} \chi_{E_i}$

as the $E_i$ are disjoint, and $\chi_{E_i} \ge 0$ pointwise.

Since all of the $E_i$ are measurable, it follows that $f^+$ is a simple function.

Also see

 * Positive Part of Simple Function is Simple Function, a natural counterpart