Pointwise Maximum of Simple Functions is Simple/Proof 1

Theorem

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

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

Then the pointwise maximum $\max \set {f, g}: X \to \R$ is also simple function.

Proof

From Pointwise Sum of Simple Functions is Simple Function:

$f + g$ is simple.

From Scalar Multiple of Simple Function is Simple Function:

$-g$ is simple.

Then, from Pointwise Sum of Simple Functions is Simple Function, we have:

$f - g$ is simple.

From Absolute Value of Simple Function is Simple Function:

$\size {f - g}$ is simple.

From Pointwise Sum of Simple Functions is Simple Function, we then have:

$\paren {f + g} + \size {f - g}$ is simple.

Finally, from Scalar Multiple of Simple Function is Simple Function, we have:

$\dfrac 1 2 \paren {\paren {f + g} + \size {f - g} }$ is simple.

By Maximum Function in terms of Absolute Value, we have:

$\ds \max \set {f, g} = \frac 1 2 \paren {\paren {f + g} + \size {f - g} }$

so:

$\max \set {f, g}$ is simple.

$\blacksquare$