Absolute Value of Simple Function is Simple Function/Proof 1

Theorem

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

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

Then $\size f: X \to \R$, the absolute value of $f$, is also a simple function.

Proof

By Sum of Positive and Negative Parts, we have:

$\size f = f^+ + f^-$

We also have that Positive Part of Simple Function is Simple Function and Negative Part of Simple Function is Simple Function.

Hence $\size f$ is a pointwise sum of simple functions.

The result follows from Pointwise Sum of Simple Functions is Simple Function.

$\blacksquare$