Positive Part of Simple Function is Simple Function

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Theorem

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

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


Then $f^+: X \to \R$, the positive 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:

$f^+ = \ds \sum_{i \mathop = 0}^n \max \set {a_i, 0} \chi_{E_i}$

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.

$\blacksquare$


Also see


Sources