Measurable Function is Simple Function iff Finite Image Set/Corollary

Corollary to Measurable Function is Simple Function iff Finite Image Set
Let $\struct {X, \Sigma}$ be a measurable space. Let $f: X \to \R$ be a measurable function.

Then $f$ has a standard representation.

Proof
Applying the main theorem to a simple function yields the representation $(1)$:


 * $\map f x = \ds \sum_{i \mathop = 1}^n y_j \map {\chi_{B_j} } x$

which is of the required form.