Absolute Value 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 $\size f: X \to \R$, the absolute value of $f$, is also a simple function.
Proof 1
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.
$\blacksquare$
Proof 2
By Simple Function has Standard Representation, $f$ has a standard representation, say:
- $(1): \quad f = \displaystyle \sum_{k \mathop = 0}^n a_k \chi_{E_k}$
Then, for all $x \in X$:
- $\left\vert{f}\right\vert \left({x}\right) = \displaystyle \left\vert{ \sum_{k \mathop = 0}^n a_k \chi_{E_k} \left({x}\right) }\right\vert$
by definition of pointwise absolute value.
The fact that $(1)$ forms a standard representation ensures that for every $x \in X$, precisely one $k$ has $x \in E_k$.
Now suppose that $x \in E_l$.
Then $\chi_{E_l} \left({x}\right) = 0$ if and only if $k \ne l$ by definition of characteristic function.
It follows that $\left\vert{f}\right\vert \left({x}\right) = \left\vert{a_l \cdot 1}\right\vert = \left\vert{a_l}\right\vert$.
Now define $g: X \to \R$ by:
$g \left({x}\right) := \displaystyle \sum_{k \mathop = 0}^n \left\vert{a_k}\right\vert \chi_{E_k} \left({x}\right)$
By construction, $g$ is a simple function, and for $x \in E_l$, $g \left({x}\right) = \left\vert{a_l}\right\vert$.
Thus, since every $x$ is in $E_l$ for precisely one $l$, it is showed that $g = \left\vert{f}\right\vert$.
As $g$ was a simple function, it follows that $\left\vert{f}\right\vert$ is, too.
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.7 \ \text{(vi)}$