Constant Function is Measurable/Proof 2

Proof
From Characteristic Function of Universe, we can write:


 * $\map f x = c \map {\chi_X} x$

for each $x \in X$.

From the definition of a $\sigma$-algebra, we have:


 * $X \in \Sigma$

So:


 * $f$ is a simple function.

Then, from Simple Function is Measurable, we have:


 * $f$ is $\Sigma$-measurable.