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

Also see

 * Operation Induced on Set of Mappings, of which the definition of $f \cdot g$ is an instantiation
 * Scalar Multiple of Simple Function is Simple Function
 * Pointwise Sum of Simple Functions is Simple Function
 * Space of Simple Functions forms Ring