Pointwise Sum of Simple Functions is Simple Function

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

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

Then $f + g: X \to \R, \left({f + g}\right) \left({x}\right) := f \left({x}\right) + g \left({x}\right)$ is also a simple function.

Proof
We have $f + g = + \circ \left\langle{f, g}\right\rangle \circ \Delta_X$, where:


 * $\Delta_X: X \to X \times X$ is the diagonal mapping on $X$
 * $\left\langle{f, g}\right\rangle: X \times X \to \R \times \R, \left\langle{f, g}\right\rangle \left({x, y}\right) := \left({f \left({x}\right), g \left({y}\right)}\right)$
 * $+: \R \times \R \to \R$ is real addition

From this, we see that $+$ may be restricted to $\operatorname{Im} \left({\left\langle{f, g}\right\rangle}\right)$, the image of $\left\langle{f, g}\right\rangle$.

It is immediate that this image equals $\operatorname{Im} \left({f}\right) \times \operatorname{Im} \left({g}\right)$.

By Real-Valued Function is Simple Function iff Finite Image Set, $\operatorname{Im} \left({f}\right)$ and $\operatorname{Im} \left({g}\right)$ are finite.

Therefore, by Cardinality of Cartesian Product, $\operatorname{Im} \left({\left\langle{f, g}\right\rangle}\right)$ is, too.

Next, from Corollary to Image of Composite Mapping, $\operatorname{Im} \left({\left\langle{f, g}\right\rangle \circ \Delta_X}\right) \subseteq\operatorname{Im} \left({\left\langle{f, g}\right\rangle}\right)$, whence the former is finite.

Now $+$ is a surjection $+: \operatorname{Im} \left({\left\langle{f, g}\right\rangle \circ \Delta_X}\right) \to \operatorname{Im} \left({f + g}\right)$ by Mapping is Surjection to Image.

By Cardinality of Surjection, this implies that $\operatorname{Im} \left({f + g}\right)$ is finite.

Whence Real-Valued Function is Simple Function iff Finite Image Set grants that $f + g$ is a simple function.

Also see

 * Operation Induced on Set of Mappings, of which the definition of $f + g$ is an instantiation
 * Scalar Multiple of Simple Function is Simple Function
 * Pointwise Product of Simple Functions is Simple Function
 * Space of Simple Functions forms Algebra