Absolute Value of Simple Function is Simple Function/Proof 1

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Theorem

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

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


Then $\left\vert{f}\right\vert: X \to \R$, the absolute value of $f$, is also a simple function.


Proof

By Sum of Positive and Negative Parts, we have:

$\left\vert{f}\right\vert = f^+ + f^-$

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

Hence $\left\vert{f}\right\vert$ is a pointwise sum of simple functions.


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

$\blacksquare$