Pointwise Sum of Simple Functions is Simple Function

Theorem

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

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

Then the pointwise sum $f + g: X \to \R$ of $f$ and $g$:

$\forall x, y \in X: \map {\paren {f + g} } x := \map f x + \map g x$

is also a simple function.

Proof

We have $f + g = + \circ \innerprod f g \circ \Delta_X$, where:

$\Delta_X: X \to X \times X$ is the diagonal mapping on $X$

$\innerprod f g: X \times X \to \R \times \R, \map {\innerprod f g} {x, y} := \tuple {\map f x, \map g y}$

$+: \R \times \R \to \R$ is real addition.

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From this, we see that $+$ may be restricted to $\Img {\innerprod f g}$, the image of $\sequence {f, g}$.

It is immediate that this image equals $\Img f \times \Img g$.

By Measurable Function is Simple Function iff Finite Image Set, $\Img f$ and $\Img g$ are finite.

Therefore, by Cardinality of Cartesian Product of Finite Sets, $\Img {\sequence {f, g} }$ is also finite.

Next, from Corollary to Image of Composite Mapping:

$\Img {\innerprod f g \circ \Delta_X} \subseteq \Img {\innerprod f g}$

whence the former is finite.

Now $+$ is a surjection:

$+: \Img {\innerprod f g \circ \Delta_X} \to \Img {f + g}$

by Restriction of Mapping to Image is Surjection.

By Cardinality of Codomain of Surjection, this implies that $\Img {f + g}$ is finite.

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

$\blacksquare$

Also see

• Definition:Pointwise Operation, 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

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.7 \ \text{(iv)}$