# Positive Part of Simple Function is Simple Function

## Theorem

Let $\left({X, \Sigma}\right)$ 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:

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

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

$f^+ = \displaystyle \sum_{i \mathop = 0}^n \max \left\{{a_i, 0}\right\} \chi_{E_i}$

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.

$\blacksquare$