Pointwise Difference of Simple Functions is Simple Function

Theorem

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

Let $f,g : X \to \R$ be simple functions.

Let $f - g: X \to \R$ be the pointwise difference of $f$ and $g$:

$\forall x \in X: \map {\paren {f - g} } x := \map f x - \map g x$

Then $f - g$ is also a simple function.

Proof

By Scalar Multiple of Simple Function is Simple Function, $-g = -1 \cdot g$ is a simple function.

By Pointwise Sum of Simple Functions is Simple Function, so is $f - g$.

$\blacksquare$